aa r X i v : . [ m a t h . OA ] A ug PRESERVATION OF ALGEBRAICITY IN FREE PROBABILITY
GREG W. ANDERSON
Abstract.
We show that any matrix-polynomial combination of free noncom-mutative random variables each having an algebraic law has again an algebraiclaw. Our result answers a question raised by a recent paper of Shlyakhtenkoand Skoufranis. The result belongs to a family of results with origins outsidefree probability theory, including a result of Aomoto asserting algebraicity ofthe Green function of random walk of quite general type on a free group.
Contents
1. Statement of the main result and introduction 12. Background for the main result and a reduction of the proof 53. Hessenberg-Toeplitz matrices and free cumulants 114. The linearization step 155. Solving the generalized Schwinger-Dyson equation 226. Notes on Newton-Puiseux series 267. Evaluation of algebraic power series on matrices 298. Proof of the main result 35References 391.
Statement of the main result and introduction
Our main result is as follows:
Theorem 1.
Let ( A , φ ) be a noncommutative probability space. Let x , . . . , x q ∈ A be freely independent noncommutative random variables. Let X ∈ Mat p ( C h x , . . . , x q i ) ⊂ Mat p ( A ) be a matrix. If the laws of x , . . . , x q are algebraic, then so is the law of X . This paper is devoted to a proof of Theorem 1. We say for short that X is a freematrix-polynomial combination of x , . . . , x q . See § Date : August 14, 2014.2010
Mathematics Subject Classification.
Key words and phrases.
Free probability, algebraicity, Schwinger-Dyson equation, lineariza-tion, realization, formal language.
The phenomenon of preservation of algebraicity established in general here haspreviously been verified in many special cases, often merely as a byproduct. Hereare several important examples. (i) It is implicit in the theory of the R -transformintroduced in [35] that free (additive) convolution preserves algebraicity. We notethat this phenomenon has been exploited in a practical way in [13]. Similar remarksapply to free multiplicative convolution. (ii) It is implicit in the theory of commuta-tors of free random variables developed in [27] that formation of free commutatorspreserves algebraicity. (iii) Algebraicity of the law of a free matrix-polynomial com-bination of semicircular variables is asserted in [31, Thm. 5.4]. This result formspart of a result of wide scope, namely [31, Thm. 1.1], which gives important con-straints on the structure of the law of a free matrix-polynomial combination ofnoncommutative random variables each of which has a nonatomic law.Our main result answers a question raised by [31].One further example of algebraicity, originating outside free probability, deservesspecial mention as the archetype for Theorem 1. Theorem 2 (See, e.g., [33, Cor. 6.7.2, p. 210]) . Let G be a free group. Let C [ G ] bethe group algebra of G with complex coefficients. Let τ : C [ G ] → C be the unique C -linear map such that τ ( g ) = δ g for g ∈ G . Then for any P ∈ C [ G ] and g ∈ G theformal power series P ∞ n =0 τ ( gP n ) t n ∈ C [[ t ]] is algebraic over the field of rationalfunctions C ( t ) . It is remarked in [22] that Theorem 2 has been frequently rediscovered. We donot know the identity of the first discoverer.Now a unitary noncommutative random variable factors as a product of twofree Bernoulli variables. In other plainer words, a group generated by elements y , . . . , y q subject only to the relations y i = 1 for i = 1 , . . . , q contains a freesubgroup on q generators, for example that generated by y y , y y , . . . , y q − y q .Thus Theorem 2 in the case g = 1 is a consequence of Theorem 1. Given this closerelationship, one reasonably looks to the proof of Theorem 2 for clues concerningthe proof of Theorem 1.The proof of Theorem 2 given in [33] is based on [15] which in turn buildsupon the theory of algebraic noncommutative formal power series (see e.g., [29] or[33, Chap. 6]), and for the latter, crucial foundations are laid in the seminal paper[11] on context free languages. Tools from the same kit are also used to prove[31, Thm. 5.4].Somewhat counterintuitively, we prove Theorem 1 by an approach mostly avoid-ing formal language theory. We rely instead on methods from free probability, ran-dom walk on groups, algebraic geometry and commutative algebra. Formal languagetheory is still involved, but in a different and simpler way. A proof of Theorem 1parallel to that of [31, Thm. 5.4] might yet be possible and would be very inter-esting. For now the sticking point seems to be that no suitable generalization of[31, Lemma 5.12] is obviously on offer. It is conceivable that our methods couldhave in the reverse direction some impact on formal language theory.Questions quite similar to that answered by Theorem 2 have been treated in theliterature of random walk on groups. Here are two particularly important examples.(i) In [4] algebraicity of the Green function of random walk of a fairly general type ona finitely generated free group was proved by explicit calculation. This is the earliestpaper of which we are aware which up to some mild and ultimately removablehypotheses proves Theorem 2. (ii) In [40] algebraicity of the Green function of any RESERVATION OF ALGEBRAICITY IN FREE PROBABILITY 3 finitely supported random walk on a group with a finitely generated free subgroupof finite index was proved by a method based on formal language theory, a methodmuch the same as later used in [33] to prove Theorem 2.The overall approach to algebraicity used here is adapted not from the literatureof formal language theory, but rather from that of random walk on groups andgraphs, especially free groups and infinite trees. See [41] for background. From therandom walk literature we cite as particular examples [4], [23] and [26]. This shortlist could be greatly extended because the idea we are adapting is a fundamentaltrope. There are two main components to this trope. Firstly, one gets relativelysimple recursions for Green functions by exploiting finiteness of cone types, or somerelated principle of self-similarity. Such recursions are often (in effect) consequencesof the familiar matrix inversion formula (cid:20) a bc d (cid:21) − = (cid:20) − (cid:21) + (cid:20) − d − c (cid:21) ( a − bd − c ) − (cid:2) − bd − (cid:3) . Here we put the Boltzmann-Fock space model of free random variables [35] intosuitably “arboreal” form in order to gain access to similar recursions. Secondly,there are criteria available for recognizing when recursions have algebraic solutions.Of course formal language theory provides such criteria, but there are also lesscomplicated criteria. For example, [23, Prop. 5.1] is an especially clear and generalcriterion, and it is applied in the cited paper in an instructive manner. We use asimilar but simpler criterion here. See Proposition 2.4.2 below.We remark that when writing [2], wherein was presented an algebraicity criterionsimilar to if rather more complicated than [23, Prop. 5.1], namely [2, Thm. 6.1], theauthors were unfortunately unaware of [23]. We wish to acknowledge the priority.We also remark that relations between algebraicity and positivity are highlydeveloped in the random walk literature, e.g., in [23] and [26], leading to local limittheorems. We do not touch those ideas here but we think they could be fruitfullyapplied in the free probability context.The paper [8] has been an important influence because, building upon operator-valued R -transform theory [37], [12] it reveals a rich algebraic and analytic structureto exploit for studying algebraicity. In particular, study of the fixed point equationsstated in [8, Thm. 2.2] should in principle lead to a proof of Theorem 1. But becausethe many-variable setting for these equations is too difficult for us to handle, wework instead with a lightly modified version of the original setup of [35] and we usejust the single classical parameter z . It remains an open and interesting problem toprove algebraicity in the many-variable setting of [8].The linearization trick which we learned from the papers [16] and [17], which werefined and used in [1], and which in the refined form was also used in [8] playshere an important role as well. But as we have recently learned from [19] andwish to acknowledge here, the trick in its refined form already exists in the litera-ture. A theorem of Sch¨utzenberger [30] (see also [7, Thm. 7.1, p. 18]) belonging toformal language theory contains the core idea of the trick sans the self-adjointness-preserving aspect. The whole trick is contained in [19, Lemma 4.1] and is calledthere symmetric realization . (There is a minor issue that the lemma is stated in thereal case but the hermitian generalization is routine.) The cited lemma appears ina context perhaps not at first glance closely connected with free probability, butclearly and closely allied with linear systems theory [10], [20]. The author planswith several co-authors to report on these interconnections in a forthcoming paper. GREG W. ANDERSON
As it happens, because Theorem 1 does not mention self-adjointness, all weneed of the linearization/realization technique is Sch¨utzenberger’s theorem and ofthe latter we in fact need only a fragment, which we prove here “from scratch”in a paragraph. See Lemma 4.5.1 below. See also § P ∈ C [ G ] with complex coefficients, one considers a square matrix P ∈ Mat n ( Z [ G ])of group-ring elements with integer coefficients, one considers not the usual gener-ating function but rather the zeta-function exp ∞ X k =1 τ (tr ( P k )) t k k ! ∈ t C [[ t ]] , and finally, one concludes strikingly that the latter both has integer coefficientsand is algebraic. Thus motivated, we raise the following question. Is “integral-algebraicity of zeta-functions” preserved under “integral-matrix-polynomial com-bination” of free random variables? Perhaps this question could be answered bycombining methods used here with those of the cited papers.As mentioned above, our formal algebraic setup for proving Theorem 1 is basedon the original setup of [35]. From that starting point, we make the rest of ourdefinitions so as to keep our approach to proving Theorem 1 as simple as possible.We make no positivity assumptions—moment sequences of variables can be arbi-trarily prescribed sequences of complex numbers. We work over the field C ((1 /z ))of formal Laurent series, using simple ideas about Banach algebras over completeultrametrically normed fields in lieu of operator theory over the complex numbers.Our method reveals nothing about the branch points of the algebraic functions itproduces. It is an open problem to recover information about positivity and branchpoints. Perhaps this is only a matter of unifying features of the several theories men-tioned above, but in our opinion some further ingredients from algebraic geometrywill be needed. The soliton theory literature, e.g., [25], might provide guidance.Here is an outline of the paper. In §
2, after filling in background in leisurely fash-ion, and in particular writing down a simple algebraicity criterion, namely Proposi-tion 2.4.2 below, we reformulate Theorem 1 as the conjunction of two propositionsboth of which concern the generalized Schwinger-Dyson equation. In § § § § § § RESERVATION OF ALGEBRAICITY IN FREE PROBABILITY 5 Background for the main result and a reduction of the proof
After recalling principal definitions, fixing notation, and filling in background forTheorem 1, we reduce Theorem 1 to two propositions each treating some aspect ofthe generalized Schwinger-Dyson equation.2.1.
Noncommutative probability spaces and free independence.
Wepresent a brief review to fix notation. See, e.g., [3], [28], [32], or [38] for background.2.1.1.
Algebras.
All algebras in this paper are unital, associative, and have a scalarfield containing C . The unit of an algebra A is denoted by 1 A ; other notation, e.g.,simply 1, may be used when context permits. Given elements x , . . . , x q ∈ A of analgebra, let C h x , . . . , x q i ⊂ A denote the subring of A generated by forming allfinite C -linear combinations of monomials in the given elements x , . . . , x q , includingthe “empty monomial” 1 A . (But if A is commutative, instead of C h x , . . . , x q i , wewrite C [ x , . . . , x q ] as is usual in commutative algebra.)2.1.2. Noncommutative probability spaces. A state φ on an algebra A is simply a C -linear functional φ : A → C such that φ (1 A ) = 1. In our formal algebraic setupno positivity constraints are imposed. A noncommutative probability space is a pair( A , φ ) consisting of an algebra A and a state φ on that algebra. Given such a pair( A , φ ), elements of A are called noncommutative random variables .2.1.3. Matrices with algebra entries.
Given an algebra A and a positive integer n , let Mat n ( A ) denote the algebra of n -by- n matrices with entries in A . Moregenerally let Mat k × ℓ ( A ) denote the space of k -by- ℓ matrices with entries in A . For A ∈ Mat k × ℓ ( C ) and a ∈ A we define A ⊗ a ∈ Mat k × ℓ ( A ) by ( A ⊗ a )( i, j ) = A ( i, j ) a .Let 1 = I n = I n ⊗ A ∈ Mat n ( A ) denote the identity matrix, as context maypermit. Let e ij ∈ Mat k × ℓ ( C ) denote the elementary matrix with 1 in position ( i, j )and 0 in every other position. Let GL n ( A ) denote the group of invertible elementsof Mat n ( A ). Given a noncommutative probability space ( A , φ ), we regard eachmatrix A ∈ Mat n ( A ) as a noncommutative random variable with respect to thestate φ n : Mat n ( A ) → C given by the formula φ n ( A ) = n P ni =1 φ ( A ( i, i )).2.1.4. Free independence.
Let ( A , φ ) be a noncommutative probability space and let A , . . . , A q ⊂ A be subalgebras such that 1 A ∈ ∩ qi =1 A i . One says that A , . . . , A q are freely independent if for every positive integer k , sequence i , . . . , i k ∈ { , . . . , q } such that i = i , i = i , . . . , i k − = i k and sequence x ∈ A i , . . . , x k ∈ A i k such that φ ( x ) = · · · = φ ( x k ) = 0, one has φ ( x · · · x k ) = 0. As a special case ofthe preceding general definition, one says that noncommutative random variables x , . . . , x q ∈ A are freely independent if the subalgebras C h x i , . . . , C h x q i ⊂ A arefreely independent.2.1.5. Univariate laws.
Let X be a variable. A univariate law (or, context permit-ting, simply a law ) is by definition a state µ : C h X i → C on the one-variablepolynomial algebra C h X i . The value µ ( X n ) ∈ C is called the n th moment of µ .Note that in our formal algebraic setup the moments of a law are allowed to bearbitrarily prescribed complex numbers. Given a noncommutative probability space( A , φ ) and a noncommutative random variable x ∈ A , the law of x is by definitionthe linear functional µ x : C h X i → C determined by the formula µ x ( X n ) = φ ( x n )for integers n ≥ GREG W. ANDERSON
Noncommutative joint laws.
Let X , . . . , X q be independent noncommutingalgebraic variables and let C h X , . . . , X q i be the noncommutative polynomial ringgenerated by these variables. A q -variable noncommutative law or, context permit-ting, simply a law , is a state on the algebra C h X , . . . , X q i . Let ( A , φ ) be a non-commutative probability space and let x , . . . , x q ∈ A be noncommutative randomvariables. The joint law µ x ,...,x q : C h X , . . . , X q i → C of the q -tuple ( x , . . . , x q )is by definition the linear functional defined by the rule µ x ,...,x q ( f ( X , . . . , X q )) = φ ( f ( x , . . . , x q )) for f ( X , . . . , X q ) ∈ C h X , . . . , X q i . The laws µ x , . . . , µ x q of theindividual variables (by analogy with classical probabilistic usage) are called the marginal laws for the joint law µ x ,...,x q . A point worth emphasizing is that if x , . . . , x q are freely independent, then the joint law µ x ,...,x q is uniquely deter-mined by the marginal laws µ x , . . . , µ x q .2.2. The Laurent series field C ((1 /z )) and related notions. We recall severaldefinitions together providing a framework in which to discuss algebraicity. See thetext [6] by Artin for background on valued fields and algebraic functions.2.2.1.
Definition of C ((1 /z )) and related objects. Let C ((1 /z )) denote the set of formal Laurent series in z of the form(1) f = X i ∈ Z c i z i ( c i ∈ C and c i = 0 for i ≫ . (The coefficients c i are not subject to any majorization.) Equipped with additionand multiplication in evident fashion, the set C ((1 /z )) becomes a field. Note thatwe have inclusions C [ z ] ⊂ C ( z ) ⊂ C ((1 /z )) and C [[1 /z ]] ⊂ C ((1 /z ))where C [ z ] is the ring of polynomials in z , C [[1 /z ]] is the ring of formal power seriesin 1 /z , and C ( z ) is the field of rational functions of z , all with coefficients in C .Note also that we have an additive direct sum decomposition C ((1 /z )) = C [ z ] ⊕ (1 /z ) C [[1 /z ]] . In our algebraic setup the formal variable z corresponds to the classical parameter z in the upper half-plane.2.2.2. Algebraic elements of C ((1 /z )) and their irreducible equations. Let C [ x, y ]be the polynomial ring over C in two independent (commuting) variables x and y .We say that f ∈ C ((1 /z )) is algebraic if one and hence all three of the followingequivalent conditions hold: • There exists some 0 = P ( x, y ) ∈ C [ x, y ] such that P ( z, f ) = 0. • There exists some 0 = Q ( x, y ) ∈ C [ x, y ] such that Q (1 /z, f ) = 0. • The field C ( z, f ) generated over C ( z ) by f is a vector space of finite dimen-sion over C ( z ).As is well-known, the algebraic elements form a subfield of C ((1 /z )) containing C ( z ). For algebraic f ∈ C ((1 /z )) there exists irreducible F ( x, y ) ∈ C [ x, y ] uniqueup to a constant multiple such that F (1 /z, f ) = 0. (The insertion of 1 /z in thepreceding definition rather than z is a technical convenience.) With but slight abuseof language we call any such irreducible polynomial the irreducible equation of f . RESERVATION OF ALGEBRAICITY IN FREE PROBABILITY 7
Valuations.
For f ∈ C ((1 /z )) expanded as on line (1) we defineval f = sup { i ∈ Z | c i = 0 } = (the valuation of f ) ∈ Z ∪ {−∞} . Note thatval f = −∞ ⇔ f = 0 , (2) val( f f ) = val f + val f , (3) val( f + f ) ≤ max(val f , val f ) with equality if val f = val f .(4)Thus val is (the logarithm of) a nonarchimedean valuation in the sense of [6]. Thusit becomes possible to use (ultra)metric space ideas to reason about C ((1 /z )) andrelated objects, as in [6], and we will do so throughout this paper. We may speakfor example of completeness. It is easy to see that C ((1 /z )) is complete with respectto the valuation val.2.2.4. Banach algebra structure for
Mat n ( C ((1 /z ))) . We now extend the metricspace ideas a bit farther. We equip the matrix algebra Mat n ( C ((1 /z ))) with a val-uation by the rule val A = max ni,j =1 val A ( i, j ). Then (2) and (4) hold for matrices,(3) holds for multiplication of a matrix by a scalar, and (3) holds for multiplicationof two matrices provided that “=” is relaxed to “ ≤ .” Thus Mat n ( C ((1 /z ))) becomesa Banach algebra over C ((1 /z )). Later, in §
3, a certain infinite-dimensional Banachalgebra over C ((1 /z )) will be introduced.2.2.5. Composition of Laurent series.
The composition f ◦ g ∈ C ((1 /z )) of f, g ∈ C ((1 /z )) is defined provided that val g >
0. The set z + C [[1 /z ]] forms agroup under composition. This group acts on the right side of C ((1 /z )) by C -linearfield automorphisms. Lemma 2.2.6. If f, g ∈ z + C [[1 /z ]] satisfy f ◦ g = z and f is algebraic, then g isalso algebraic.Proof. For some 0 = P ( x, y ) ∈ C [ x, y ] we have 0 = P ( z, f ) ◦ g = P ( g, z ). (cid:3) Algebraicity of univariate laws.
We recall how to attach to each univari-ate law a (formal) Stieltjes transform and a (modified formal) R -transform `a la Voiculescu. Then we recall how in terms of these transforms one can characterizealgebraicity of a law.2.3.1.
Formal Stieltjes transforms.
For a law µ : C h X i → C , the formal sum S µ ( z ) = ∞ X n =0 µ ( X n ) /z n +1 ∈ C ((1 /z ))is by definition the formal Stieltjes transform of µ . Hereafter we drop the adjective“formal” since no other kind of Stieltjes transform will be considered in this paper.2.3.2. Algebraicity of univariate laws.
A law µ will be called algebraic if its Stieltjestransform S µ ( z ) ∈ C ((1 /z )) is algebraic. GREG W. ANDERSON
Free cumulants and R -transforms. Given a law µ : C h X i → C one defines infree probability theory for each positive integer n the n th free cumulant κ n ( µ ) ∈ C .This can be done various ways, e.g., with generating functions or combinatoriallyusing noncrossing partitions. See, e.g., [3], [28], or [38] for background; the foundingdocument for this theory is [35]. The generating function R µ ( t ) = ∞ X n =1 κ n ( µ ) t n − ∈ C [[ t ]]for the free cumulants is the formal version of the R -transform of Voiculescu. Here-after we drop the adjective “formal” since no other kind of R -transform will beconsidered in this paper.2.3.4. Modified R -transforms. To define and make calculations with free cumulants,we will use the generating function method. Consider the modified R -transform˜ R µ ( z ) = z + R µ (1 /z ) = z + ∞ X n =1 κ n ( µ ) z − n ∈ z + C [[1 /z ]] , which we will find slightly more convenient. Obviously each of R µ ( t ) and ˜ R µ ( z )uniquely determines the other. It is known (see [3], [28], [35], or [38]) that ˜ R µ ( z ) isthe unique solution of the equation(5) (cid:18) S µ ( z ) (cid:19) ◦ e R µ ( z ) = z. (cid:18) Equivalently: e R µ ( z ) ◦ (cid:18) S µ ( z ) (cid:19) = z. (cid:19) Since z + C [[1 /z ]] is a group under composition, the modified R -transform e R µ ( z )is well-defined for every law µ , hence the sequence { κ n ( µ ) } ∞ n =1 of free cumulants isdefined, and it uniquely determines µ . Furthermore the free cumulants of a law canbe arbitrarily prescribed.The next lemma expresses algebraicity in terms of free cumulants. Lemma 2.3.5.
Let µ : C h X i → C be a law. Then the following statements areequivalent: (I) µ is algebraic. (II) S µ ( z ) is algebraic. (III) e R µ ( z ) is algebraic.Proof. The equivalence (I) ⇔ (II) holds by definition. The equivalence (II) ⇔ (III)holds by Lemma 2.2.6, statement (5) above, and the fact that 0 = f ∈ C ((1 /z )) isalgebraic if and only if the reciprocal 1 /f is algebraic. (cid:3) An algebraicity criterion.
We now present the algebraicity criterion whichwe will use to take the final step of the proof of Theorem 1. To do so we abruptlyswitch to the optic of commutative algebra.2.4.1.
Setup for the criterion.
Let
K/K be any extension of fields. Let x = ( x , . . . , x n ) be an n -tuple of independent (commuting) variables and let K [ x ]be the polynomial ring generated over K by these variables. Let f = ( f , . . . , f n ) = f ( x ) ∈ K [ x ] n be an n -tuple of polynomials. Let J ( x ) = det ni,j =1 ∂f i ∂x j ∈ K [ x ] bethe determinant of the Jacobian matrix of f . Let α = ( α , . . . , α n ) ∈ K n be an n -tuple such that f ( α ) = 0 but J ( α ) = 0. RESERVATION OF ALGEBRAICITY IN FREE PROBABILITY 9
Proposition 2.4.2.
Notation and assumptions are as above. Every entry of thevector α is algebraic over K . This statement is the same as [24, Prop. 5.3, Chap. VIII, p. 371], and also thesame as [33, Lemma 6.6.9, Chap. 6, p. 206].2.5.
Large-scale organization of the proof of Theorem 1.
We recall the gen-eralized Schwinger-Dyson equation and then we state two technical results aboutit which together imply Theorem 1.2.5.1.
The generalized Schwinger-Dyson equation.
In the two technical propositionsto be formulated below we consider an instance(6) I n + a (0) g + q X θ =1 ∞ X j =2 κ ( θ ) j ( a ( θ ) g ) j = 0of the generalized Schwinger-Dyson equation for which the data are(7) positive integers q and n ,matrices g, a (0) ∈ Mat n ( C ((1 /z ))),matrices a (1) , . . . , a ( q ) ∈ Mat n ( C ), anda family (cid:26)n κ ( θ ) j o ∞ j =2 (cid:27) qθ =1 of complex numbers.We assume that(8) lim j →∞ val ( a ( θ ) g ) j = −∞ for θ = 1 , . . . , q in order that the left side of (6) have a well-defined value in Mat n ( C ((1 /z ))). Fur-thermore, we impose the following nondegeneracy condition:The linear map(9) h a (0) h + q X θ =1 ∞ X j =2 j − X ν =0 κ ( θ ) j ( a ( θ ) g ) ν ( a ( θ ) h )( a ( θ ) g ) j − − ν : Mat n ( C ((1 /z ))) → Mat n ( C ((1 /z ))) is invertible.Note that the map above is well-defined by assumption (8).We will prove the following two results. Proposition 2.5.2.
Let ( A , φ ) be a noncommutative probability space. Let x , . . . , x q ∈ A be freely independent noncommutative random variables. Let X ∈ Mat p ( C h x , . . . , x q i ) ⊂ Mat p ( A ) be a matrix. (Up to this point we have merely repeated the setup for Theorem 1.) Forindices θ = 1 , . . . , q and j = 2 , , , . . . , let κ ( θ ) j denote the j th free cumulant of thelaw of the noncommutative random variable x θ . Then the family nn κ ( θ ) j o qθ =1 o ∞ j =2 of complex numbers for some integer n > p can be completed to a family (cid:18) q, n, g, a (0) , n a ( θ ) o qθ =1 , (cid:26)n κ ( θ ) j o ∞ j =2 (cid:27) qθ =1 (cid:19) of the form (7) satisfying (6) , (8) , and (9) along with the further conditions a (0) ∈ { A + Bz | A, B ∈ Mat n ( C ) } and (10) S µ X = − p p X i =1 g ( i, i ) . (11) Proposition 2.5.3.
Let data of the form (7) satisfy (6) , (8) , and (9) . Assumefurthermore that a (0) ∈ Mat n ( C ( z )) and (12) P ∞ j =2 κ ( θ ) j +1 z − j ∈ C ((1 /z )) is algebraic for θ = 1 , . . . , q . (13) Then every entry of the matrix g is algebraic. See § § § §
7, and § Reduction of the proof of Theorem 1.
In view of Lemma 2.3.5, it is clear thatPropositions 2.5.2 and 2.5.3 together imply Theorem 1.2.5.5.
Remark.
In the simple case X = x + · · · + x q ∈ C h x , . . . , x q i ⊂ Mat ( A ) = A , the instance of generalized Schwinger-Dyson equation emerging from the proof ofProposition 2.5.2 reduces to the standard fact [35] that the R -transform is additivefor the addition of free random variables.2.5.6. Remark.
The (un)generalized Schwinger-Dyson equation is familiar in thecase that κ ( θ ) j = κ j ( µ x θ ) = 0 for j >
2. In the latter special case the equation (6)arises naturally in the study of free matrix-polynomial combinations of semicircularvariables. See, e.g., [1], [3], [8], [16], [17], [18], [28] and [38].2.5.7.
Remark.
Proposition 2.5.2 is unsurprising. It is proved here by straightfor-wardly combining three standard methodologies, namely:(i) the Boltzmann-Fock space model of free random variables,(ii) the linearization/realization method, and(iii) recursions of a type occurring in the study of random walk on infinite trees.Methodology (i) clearly originates in [35]. Methodology (ii) we learned from thepapers [16] and [17], and subsequently we refined it in [1], but as we have recentlylearned from [19], the essential point apart from the self-adjointness-preservingaspect is contained in Sch¨utzenberger’s theorem [7, Thm. 7.1, p. 18]. Methodology(iii) has an obscure origin since random walk on many types of graphs has beenstudied in probability theory for decades and many methods for getting recursionshave become commonplace. In this case we point to the examples [4], [23], [26] and[40] as inspirations, and refer the reader to [41] for background.2.5.8.
Remark.
The form of Proposition 2.5.2 is in key respects quite similar to thatof [8, Thm 2.2]. Indeed it could not be essentially different since it has the sameorigins in operator-valued free probability theory. In particular, (6) can be rewrittenas a fixed point equation. But the matrix upper half-plane plays no role either inthe statement or the proof of Proposition 2.5.2, greatly simplifying matters.
RESERVATION OF ALGEBRAICITY IN FREE PROBABILITY 11
Remark.
In the semicircular case κ ( θ ) j = κ j ( µ x θ ) = 0 for j > κ ( θ ) j = κ j ( µ x θ ) = 0 for all but finitely many pairs ( j, θ ).2.5.10. Remark.
We continue in the setup of the preceding remark. Consider nowthe remaining case in which κ ( θ ) j = κ j ( µ x θ ) = 0 for infinitely many pairs ( j, θ ).Then one can no longer prove Proposition 2.5.3 by applying Proposition 2.4.2 di-rectly to solutions of the system of equations (6) because this system no longerconsists exclusively of polynomial equations. This obstruction is the main difficultyof the proof of Theorem 1. We overcome the obstruction by means of the theory ofNewton polygons (see § § n + 3 qn polynomialequations in n + 3 qn unknowns which in a useful sense contains the system (6) andto which Proposition 2.4.2 may be applied directly. (See § Hessenberg-Toeplitz matrices and free cumulants
We introduce the formal version of operator theory used in this paper and then asan illustration we revisit a key insight of Voiculescu concerning the free cumulants.3.1.
The algebras M and M ((1 /z )) . We introduce two algebras of infinite ma-trices, the first an algebra over C and the second a larger algebra over C ((1 /z ))possessing Banach algebra structure.3.1.1. Notation.
Let N denote the set of nonnegative integers.3.1.2. The algebra M . Let M denote the vector space over C consisting of N -by- N matrices M such that for each j ∈ N there exist only finitely many i ∈ N such that M ( i, j ) = 0. Note that for each M ∈ M the entry in the upper left corneris denoted by M (0 , M (1 , ∈ N . Note that every upper-triangular N -by- N matrix with entries in C belongs to M . Informally, M consists of the “almost upper-triangular” matrices. Itis easy to see that matrix multiplication is well-defined on M and moreover asso-ciative, thus making M into a unital associative algebra with scalar field C . Indeed,it is clear that M is a copy of the algebra of linear endomorphisms of a complexvector space of countably infinite dimension. We write = 1 M to abbreviate nota-tion. We equip M with the state φ ( M ) = M (0 , M , φ ).3.1.3. The algebra M ((1 /z )) . Let M ((1 /z )) denote the set of N -by- N matrices M with entries in C ((1 /z )) satisfying one and hence both of the following equivalentconditions: • There exists a
Laurent expansion M = P n ∈ Z M n z n with coefficients M n ∈ M such that M n = 0 for n ≫ • One has lim i →∞ val M ( i, j ) = −∞ for each j ∈ N (without any requirementof uniformity in j ) and furthermore one has sup i,j ∈ N val M ( i, j ) < ∞ . From the equivalent points of view described above it is clear that M ((1 /z )) be-comes a unital C ((1 /z ))-algebra with respect to the usual notion of matrix multipli-cation. For M ∈ M ((1 /z )) we define val M = sup i,j ∈ N val M ( i, j ). With respect tothe valuation function val thus extended to M ((1 /z )), the latter becomes a unitalBanach algebra over C ((1 /z )). We write = 1 M = 1 M ((1 /z )) .3.1.4. Elementary matrices and an abuse of notation.
Let e [ i, j ] ∈ M denote theelementary matrix with entries given by the rule e [ i, j ]( k, ℓ ) = δ ik δ jℓ for i, j, k, ℓ ∈ N .The notation e [ i, j ] ∈ M introduced here is intended to contrast with the notation e ij ∈ Mat k × ℓ ( C ) previously introduced for elementary matrices with finitely manyrows and columns. For M ∈ M supported in a set S ⊂ N × N which intersects eachcolumn N × { j } in a finite set, we abuse notation by writing M = X ( i,j ) ∈ S M ( i, j ) e [ i, j ]as a shorthand to indicate the placement of entries of M .The following simple lemma is a key motivation for the definition of M ((1 /z )). Lemma 3.1.5.
Fix M ∈ M arbitrarily and let µ denote the law of M . Then thematrix z − M ∈ M ((1 /z )) is invertible and S µ ( z ) = ( z − M ) − (0 , .Proof. One has ( z − M ) − = 1 z ∞ X k =0 M k z k ∈ M ((1 /z )) . The geometric series here is convergent because val Mz <
0. This noted, it is clearthat the series ( z − M ) − (0 ,
0) is the Stieltjes transform of the law of M . (cid:3) Remark.
Our setup is inspired by (but is much simpler than) that of [5], andthus belongs to the lineage of [34]. The theory of the R -transform overlaps in aninteresting way with the theory of residues developed in [34], one point of contactbeing the notion of a Hessenberg-Toeplitz matrix. (See immediately below.) Thisconnection deserves further investigation.3.2. Hessenberg-Toeplitz matrices.
Basic definitions.
Let { κ j } ∞ j =1 be any sequence of complex numbers. Con-sider the infinite matrix(14) H = κ κ κ . . . κ κ κ . . . κ κ κ . . . κ κ κ . . . . . . . . . . . . . . . ∈ M . Equivalently, in terms of the elementary matrices e [ i, j ] ∈ M we have(15) H = X k ∈ N e [1 + k, k ] + X j ∈ N κ j +1 e [ k, j + k ] . RESERVATION OF ALGEBRAICITY IN FREE PROBABILITY 13
The matrix H displays the (upper) Hessenberg pattern: i > j + 1 ⇒ H ( i, j ) = 0 for i, j ∈ N . The matrix H also displays the Toeplitz pattern: H ( i + 1 , j + 1) = H ( i, j )for i, j ∈ N . Accordingly we call H a Hessenberg-Toeplitz matrix .The reason for our interest in the matrix H is explained by the next lemma. Lemma 3.2.2.
Assumptions and notation are as above. Then for every positiveinteger j , the j th free cumulant of H viewed as a noncommutative random variablein the noncommutative probability space ( M , φ ) equals κ j . This fact is well-known—it is a key insight for Voiculescu’s theory of the R -transform [35]. It is therefore not necessary to give a proof. But we neverthe-less give a proof in § § Remark.
If one patiently works through the definitions and uses the theoryof residues from [34], one can see that Lemma 3.2.2 is equivalent to the one-variableLagrange Inversion Formula.3.3.
Inversion of block-decomposed matrices.
We pause to review a methodof calculation used repeatedly in the sequel.3.3.1.
Basic identities.
Let (cid:20) a bc d (cid:21) be an invertible square matrix (in practice infinite) decomposed into blocks where a and d are square and d is also invertible. Then we have a factorization(16) (cid:20) a − bd − c 00 d (cid:21) = (cid:20) − bd − (cid:21) (cid:20) a bc d (cid:21) (cid:20) − d − c 1 (cid:21) from which in particular we infer that the Schur complement a − bd − c is invertible.Let g = ( a − bd − c ) − . From (16) one then straightforwardly derives the inversion formula (cid:20) a bc d (cid:21) − = (cid:20) − (cid:21) + (cid:20) − d − c (cid:21) g (cid:2) − bd − (cid:3) . (17)The latter formula also shows that invertibility of d and a − bd − c implies invert-ibility of (cid:20) a bc d (cid:21) . For convenient application in §
5, we restate in abstract form acouple of relations among blocks recorded in formula (17).
Lemma 3.3.2.
Let A be a unital associative algebra (perhaps not commutative).Let π, σ ∈ A satisfy π = π = 0 , σ = σ = 0 , πσ = σπ = 0 and A = π + σ . Let A ∈ A be invertible. Assume furthermore that σAσ is invertible in the algebra σ A σ and let A − σ denote the inverse. Then we have σA − π = − A − σ AπA − π and (18) A − σ = (1 A − A − πAσ ) A − σ . (19) Proof.
We have σAσA − π = − σAA − π + σAσA − π = − σAπA − π. Now left-multiply extreme terms by A − σ to recover (18). Similarly, we have σ = σAA − σ = ( A − πAσ ) A − σ . Now left-multiply extreme terms by A − to recover (19). (cid:3) Proof of Lemma 3.2.2.
Consider the Laurent series f = f ( z ) = z + ∞ X j =1 κ j z j − ∈ z + C [[1 /z ]] . It will suffice to show that f is equal to the modified R -transform of the law of H .Consider also the Stieltjes transform g = g ( z ) = S µ H ( z ) ∈ (1 /z ) + (1 /z ) C [[1 /z ]]of the law of H . Since z + C [[1 /z ]] forms a group under composition, it will sufficeto show that z = f ◦ g , equivalently z = g − + P ∞ j =1 κ j g j − , or equivalently1 = ( z − κ ) g − ∞ X j =2 κ j g j . Let A = z − H ∈ M ((1 /z )) . By Lemma 3.1.5 the inverse G = A − ∈ M ((1 /z ))exists and furthermore g = G (0 , . In view of the relation 1 = X k ∈ N A (0 , k ) G ( k, G = A − , it will be enough simply to prove that(20) G ( i,
0) = g i +1 for i ∈ N .Now with an eye toward applying (17) above, consider the block decomposition A = (cid:20) a bc d (cid:21) where a = z − κ , b = − (cid:2) κ κ . . . (cid:3) , c = − , and d = A. By (17) we have G (1 , G (2 , = − d − cg = G (0 , G (1 , g, whence (20). The proof is complete. (cid:3) RESERVATION OF ALGEBRAICITY IN FREE PROBABILITY 15 The linearization step
In this section we apply (a fragment of) Sch¨utzenberger’s Theorem to a suitablemodel of free noncommutative random variables with prescribed free cumulants,thus advancing the proof of Proposition 2.5.2.4.1.
Stars and diamonds.
We build a model for the free unital associative monoidon q generators for which N is the underlying set. Using this monoid structure wewill be able to construct and manipulate usefully patterned matrices in M ((1 /z )).4.1.1. Notation.
The parameter q figuring in Theorem 1 is considered fixed through-out the remainder of the paper. Many constructs below depend on q but the notationfrequently does not show this.4.1.2. Improper representations to the base q . Suppose at first that q >
1. Ingrade school one learns to represent nonnegative integers to the base q using place-notation and digits selected from the set { , . . . , q − } . It is not hard to see thatusing instead digits selected from the set { , . . . , q } one still gets a unique repre-sentation for every member of N , it being understood that 0 is represented by theempty digit string ∅ . A representation to the base q of a nonnegative integer usingdigits { , . . . , q } will be called improper . Improper representations to the base q make sense also for q = 1. In the latter extreme case each x ∈ N is represented bya string of 1’s of length x .4.1.3. Example: counting improperly to the base . ∅ , , , , , , , , , , , , , , , , , , . . . The binary operation ⋆ q . We define the binary operation ⋆ = ⋆ q : N × N → N by the rule x ⋆ y = xq ℓ + y where ℓ ∈ N satisfies q ℓ − q − ≤ y ≤ q · (cid:18) q ℓ − q − (cid:19) . Informally, ℓ is the number of digits in the improper base q representation of y , and x ⋆ y is the concatenation of the digit strings of x and of y . The operation ⋆ makes N into a free associative monoid freely generated by the digits 1 , . . . , q with 0 asthe identity element. Lemma 4.1.5. N \ { } is the disjoint union of the sets N ⋆ θ for θ = 1 , . . . , q . There is nothing to prove. We record this for convenient reference since howevertrivial, this is an important point in a later proof.4.1.6.
The binary operation ⋄ q . We define the binary operation ⋄ = ⋄ q : N × N → N by the formula x ⋄ y = x ⋆ · · · ⋆ x | {z } y . We use exponential-style notation to emphasize the analogy with exponentiation inthe usual sense.
Lemma 4.1.7.
For θ = 1 , . . . , q , every x ∈ N has a unique factorization x = θ ⋄ i ⋆ k where i ∈ N and k ∈ N \ θ ⋆ N . Again, there is nothing to prove. We record this for convenient reference.4.1.8.
Remark.
Consider the graph Γ = Γ q with vertex set N and an edge connect-ing x to θ ⋆ x for each pair ( θ, x ) ∈ { , . . . , q } × N . With 0 ∈ N designated as theroot, the resulting graph Γ is an infinite rooted planar tree in which every vertexhas a “birth-ordered” set of q children, i.e., a q -ary rooted tree. We do not explicitlyuse the q -ary tree in this paper because we instead rely on the monoid ( N , ⋆ ) to doour bookkeeping. Nonetheless the notion of the q -ary tree remains a strong guideto intuition.4.2. Free random variables with prescribed free cumulants.
The next propo-sition exhibits a model for q free noncommutative random variables with prescribedfree cumulants. The model is essentially the same as that used in [35], but with thenotation designed to make recursions easy to see. Proposition 4.2.1.
Let (cid:26)n κ ( θ ) j o ∞ j =1 (cid:27) qθ =1 be any family of complex numbers. Then the family (21) X k ∈ N e [ θ ⋆ k, k ] + X j ∈ N X k ∈ N κ ( θ ) j +1 e [ k, θ ⋄ j ⋆ k ] ∈ M for θ = 1 , . . . , q of noncommutative random variables is freely independent and moreover the j th free cumulant of the θ th noncommutative random variable equals κ ( θ ) j . The proof requires some preparation and is completed in § Self-embeddings of M . For θ = 1 , . . . q and A ∈ M we define(22) A ( θ ) = X k ∈ N \ θ⋆ N X i,j ∈ N A ( i, j ) e [ θ ⋄ i ⋆ k, θ ⋄ j ⋆ k ] . By Lemma 4.1.7 the matrix A ( θ ) is block-diagonal with copies of A indexed by N \ θ ⋆ N repeated along the diagonal. Thus the map ( A A ( θ ) ) : M → M is aunital one-to-one homomorphism of algebras. Note that A ( θ ) (0 ,
0) = A (0 ,
0) andhence the map A A ( θ ) is law-preserving. Let M ( θ ) denote the embedded imageof M under the map A A ( θ ) . Lemma 4.2.3.
The subalgebras M (1) , . . . , M ( q ) ⊂ M are freely independent.Proof. Fix θ , . . . , θ k ∈ { , . . . , q } such that θ = θ , θ = θ , . . . , θ k − = θ k . Fix A , . . . , A k ∈ M such that A (0 ,
0) = · · · = A k (0 ,
0) = 0 . Our task is to verify that X ( i ,...,i k − ) ∈ N k − A ( θ )1 (0 , i ) A ( θ )2 ( i , i ) · · · A ( θ k − ) k − ( i k − , i k − ) A ( θ k ) k ( i k − , A ( θ )1 · · · A ( θ k ) k )(0 ,
0) = 0 . RESERVATION OF ALGEBRAICITY IN FREE PROBABILITY 17
Now by definition, for any matrix A ∈ M such that A (0 ,
0) = 0 and θ = 1 , . . . , q , thematrix entry A ( θ ) ( i, j ) vanishes unless i ∈ θ ⋆ N or j ∈ θ ⋆ N . It follows that for any( k − i , . . . , i k − ) ∈ N k − the corresponding term in the sum on the leftside of (23) vanishes. Thus, a fortiori , (23) holds. (cid:3) Proof of Proposition 4.2.1.
Let H θ be a copy of the matrix H defined in (14)and equivalently in (15), with κ j replaced by κ ( θ ) j . By Lemma 3.2.2 we know alreadythat the j th free cumulant of the law of H θ equals κ ( θ ) j . Substituting directly intothe definition (22) we have H ( θ ) θ = X i ∈ N X k ∈ N \ θ⋆ N e [ θ ⋄ ( i +1) ⋆ k, θ ⋄ i ⋆ k ] + X i ∈ N X j ∈ N X k ∈ N \ θ⋆ N κ ( θ ) j +1 e [ θ ⋄ i ⋆ k, θ ⋄ ( i + j ) ⋆ k ] . The result follows now via Lemmas 4.1.7 and 4.2.3. (cid:3)
Remark.
Voiculescu [35] introduced the Boltzmann-Fock space model of freerandom variables using lowering and raising operators for his striking proof of addi-tivity of the R -transform for addition of free random variables. Also see[3, Cor. 5.3.23] and its proof for a quick review of this material. Proposition 4.2.1is merely a description of the Boltzmann-Fock space model using notation chosento make recursions more easily accessible. In the setup of Proposition 4.2.1 thematrices(24) ˆ λ ( θ ) = X k ∈ N e [ θ ⋆ k, k ] ∈ M and λ ( θ ) = X k ∈ N e [ k, θ ⋆ k ] ∈ M for θ = 1 , . . . , q correspond to the lowering and raising operators considered in [35], respectively.Note that using the operators on line (24) we can rewrite the operators on (21) inthe more familiar form(25) ˆ λ ( θ ) + X j ∈ N κ ( θ ) j +1 ( λ ( θ ) ) j ∈ M for θ = 1 , . . . , q .The infinite sum here is an abuse of notation but it nonetheless makes sense becausethe matrices being summed have disjoint supports only finitely many of which meetany given column. Later we will also consider lowering and raising operators(26) ˆ ρ ( θ ) = X k ∈ N e [ k ⋆ θ, k ] ∈ M and ρ ( θ ) = X k ∈ N e [ k, k ⋆ θ ] ∈ M for θ = 1 , . . . , q acting (so to speak) on the right rather than the left. The relations λ ( θ ) ρ ( θ ′ ) = ρ ( θ ′ ) λ ( θ ) , ˆ λ ( θ ) ˆ ρ ( θ ′ ) = ˆ ρ ( θ ′ ) ˆ λ ( θ ) , (27) ρ ( θ ′ ) ˆ ρ ( θ ) = δ θθ ′ , and q X α =1 ˆ ρ ( α ) ρ ( α ) = ∞ X i =1 e [ i, i ](28)for θ, θ ′ = 1 , . . . , q are easy to verify. See for example [1, Sec. 3.4] where these andfurther relations are written out as part of an analysis leading (without any refer-ence to noncrossing partitions) to the Schwinger-Dyson equation for semicircularvariables.4.2.6. Remark.
The interplay of left and right lowering and raising operators is afundamental feature of the recently introduced bi-free framework of [36].4.3.
Kronecker products and the isomorphism ♮ . We introduce notationwhich is rather tedious to define but convenient to calculate with.
Classical Kronecker products.
Recall that for matrices of finite size the
Kro-necker product A (1) ⊗ A (2) ∈ Mat k k × ℓ ℓ ( C ) (cid:16) A ( α ) ∈ Mat k α × ℓ α ( C ) for α = 1 , (cid:17) is defined by the rule A (1) ⊗ A (2) = A (1) (1 , A (2) . . . A (1) (1 , ℓ ) A (2) ... ... A (1) ( k , A (2) . . . A (1) ( k , ℓ ) A (2) or equivalently and more explicitly (if more cumbersomely) A (1) ⊗ A (2) ( n ( i −
1) + i , n ( j −
1) + j ) = A (1) ( i , j ) A (2) ( i , j )for α = 1 , i α = 1 , . . . , k α , and j α = 1 , . . . , ℓ α .4.3.2. Kronecker products involving infinite matrices.
In the mixed infinite/finitecase we define the
Kronecker product x ⊗ a ∈ M ((1 /z )) ( x ∈ M ((1 /z )) and a ∈ Mat n ( C ((1 /z ))))by the rule x ⊗ a = x (0 , a x (0 , a . . .x (1 , a x (1 , a . . . ... ... . . . or equivalently and more explicitly( x ⊗ a )( i n + i − , j n + j −
1) = x ( i , j ) a ( i , j )for i , j ∈ N and i , j = 1 , . . . , n .We also define a ⊗ x ∈ Mat n ( M ((1 /z ))) ( a ∈ Mat n ( C ((1 /z ))) and x ∈ M ((1 /z )))by the somewhat ungainly iterated index formula(( a ⊗ x )( i , j ))( i , j ) = a ( i , j ) x ( i , j )for i , j ∈ N and i , j = 1 , . . . , n .4.3.3. The operation ♮ . For M ∈ Mat n ( M ((1 /z ))) we define M ♮ ∈ M ((1 /z )) by theformula M ♮ = X i ,j ∈ N n X i ,j =1 ( M ( i , j )( i , j ))( e [ i , j ] ⊗ e i j ) , thus defining an isometric isomorphism( M M ♮ ) : Mat n ( M ((1 /z ))) → M ((1 /z ))of Banach algebras over C ((1 /z )), where the source algebra is given Banach algebrastructure by declaring that val A = max ni,j =1 val A ( i, j ) for A ∈ Mat n ( M ((1 /z ))).Finally, note that(29) ( a ⊗ x ) ♮ = x ⊗ a for a ∈ Mat n ( C ((1 /z ))) and x ∈ M ((1 /z )) . Thus the operation ♮ has a natural interpretation as exchange of tensor factors. RESERVATION OF ALGEBRAICITY IN FREE PROBABILITY 19
Lemma 4.3.4.
Fix A ∈ Mat n ( M ) . Then the following statements hold: zI n ⊗ − A ∈ Mat n ( M ((1 /z ))) and z − A ♮ ∈ M ((1 /z )) are invertible. (cid:0) ( zI n ⊗ − A ) − (cid:1) ♮ = ( z − A ♮ ) − .S µ A ( z ) = 1 n n − X i =0 ( z − A ♮ ) − ( i, i ) = n n X i =1 ( zI n ⊗ − A ) − ( i, i ) ! (0 , . This statement supplements Lemma 3.1.5 only by some minor bookkeeping de-tails. We therefore omit proof.4.4.
Digital linearization.
Here is the main result in this section.
Proposition 4.4.1.
Let ( A , φ ) be a noncommutative probability space. Let x , . . . , x q ∈ A be freely independent noncommutative random variables. Fix X ∈ Mat p ( C h x , . . . , x q i ) ⊂ Mat p ( A ) . Let κ ( θ ) j = κ j ( µ x θ ) for j = 1 , , . . . and θ = 1 , . . . , q . Then there exist for some N > matrices L , L , . . . , L q ∈ Mat p + N ( C ) with the following properties: L vanishes identically in the upper left p -by- p block. (30) L , . . . , L q are supported in the lower right N -by- N block. (31) L = ⊗ (cid:18) L + (cid:20) zI p
00 0 (cid:21)(cid:19) + q X θ =1 X k ∈ N e [ θ ⋆ k, k ] ⊗ L θ (32) + q X θ =1 X j ∈ N X k ∈ N κ ( θ ) j +1 e [ k, θ ⋄ j ⋆ k ] ⊗ L θ ∈ M ((1 /z )) is invertible. S µ X ( z ) = 1 p p − X i =0 L − ( i, i ) . (33)We call L a digital linearization of X . It is worth remarking that this linearizationis thoroughgoing in the sense that not only do the variables x , . . . , x q appearlinearly—so also does the variable z . The proof will be completed in § Remark.
Picking up again on the idea mentioned in § L describes a random walk on the q -ary treeΓ q such that from a given vertex x ∈ N , one may (i) step one unit back to-ward the root (if not already at the root), (ii) stay in place, or (iii) step awayfrom the root arbitrarily far along along a geodesic { θ ⋄ i ⋆ x | i ∈ N } for some θ ∈ { , . . . , q } . Whether or not this interpretation of L is absurd, it is does makerandom walk intuition available to analyze L . Guided by this intuition we will provein § L , as well as for more general infinite matrices.4.5. Sch¨utzenberger’s theorem.
The next lemma recalls what we need of theself-adjoint linearization trick, and as we have already noted in the introduction,what we need boils down to Sch¨utzenberger’s Theorem [7, Thm. 7.1]. In fact weneed only a quite specialized consequence of this theorem, or rather of its proof,simple enough to prove quickly from scratch, as follows.
Lemma 4.5.1.
For each f ∈ Mat p ( C h X , . . . , X q i ) there exists a factorization f = bd − c (called a linearization of f ) where b ∈ Mat p × N ( C ) , c ∈ Mat N × p ( C ) , d ∈ GL N ( C h X , . . . , X q i ) , and each entry of d belongs to the C -linear span of , X , . . . , X q . Note that the proof below actually produces d with the further property that d − I N is strictly upper triangular. Proof.
If every entry of f belongs to the C -linear span of 1 , X , . . . , X q , then, say, f = (cid:2) I p (cid:3) (cid:20) I p − f I p (cid:21) − (cid:20) I p (cid:21) is a linearization. Thus it will be enough to demonstrate that given linearizable f , f ∈ Mat p ( C h X , . . . , X q i ), again f + f and f f are linearizable. So supposethat f i = b i d − i c i for i = 1 , f + f = (cid:2) b b (cid:3) (cid:20) d d (cid:21) − (cid:20) c c (cid:21) and f f = (cid:2) b (cid:3) d c
00 1 b d − c . To assist the reader in checking the second formula, we note that d c
00 1 b d − = d − − d − c d − c b d − − b d − d − . Thus f + f and f f have linearizations. Consequently the lemma does indeedhold. (cid:3) Remark.
We refer the reader to the book [7] for a complete discussion ofSch¨utzenberger’s Theorem and its context in the theory of rational formal noncom-mutative power series. Nonetheless, we feel that we do owe the reader at least abrief sketch of the interpretation of Lemma 4.5.1 that identifies it as a consequenceof Sch¨utzenberger’s Theorem. For simplicity and with some loss of generality weassume that p = 1. Without further loss of generality we may assume that d (0) = I N − q X θ =1 h θ ⊗ X θ . Let h = ( h , . . . , h q ) ∈ Mat qN . Then we have d − = X monomials M ∈ C h X i M ( h ) ⊗ M (the sum is actually finite on account of the remark immediately following thestatement of Lemma 4.5.1) and hence f = bd − c = X monomials M ∈ C h X i ( b M ( h ) c ) M . RESERVATION OF ALGEBRAICITY IN FREE PROBABILITY 21
Write f = X monomials M ∈ C h X i a M M ( a M ∈ C ) . In the spirit of Sch¨utzenberger’s theorem we should think of the collection( b, h , . . . , h q , c )as a “linear automaton” which by the rule a M = b M ( h ) c for monomials M ∈ C h M i “computes” f coefficient-by-coefficient. More generally and analogously,Sch¨utzenberger’s theorem produces a linear automaton to compute the coefficientsof any given rational noncommutative formal power series, and moreover assertsthat any noncommutative formal power series so “computable” is rational. Wenote finally that our application of Lemma 4.5.1 below is not dependent on its “au-tomatic” interpretation. Neither the theory of automata nor the theory of rationalformal noncommutative power series are needed in the sequel.4.6. Proof of Proposition 4.4.1.
Without loss of generality we may assume that( A , φ ) is the noncommutative probability space ( M , φ ) and we may take { x θ } qθ =1 to be the family constructed in Proposition 4.2.1. Fix f = f ( X , . . . , X q ) ∈ Mat p ( C h X , . . . , X q i )such that f ( x , . . . , x q ) = X , write f = bd − c as in Lemma 4.5.1, and then write (cid:20) bc d (cid:21) = L ⊗ C h X ,..., X q i + L ⊗ X + · · · + L q ⊗ X q ∈ Mat p + N ( C h X , . . . , X q i ) ( L , L , . . . , L q ∈ Mat p + N ( C ))in the unique possible way. Finally, let L = ⊗ (cid:18) L + (cid:20) zI p
00 0 (cid:21)(cid:19) + x ⊗ L + · · · + x q ⊗ L q ∈ M ((1 /z ))noting that this expression when expanded in terms of elementary matrices takesby (21) the desired form (32). Let B = b ⊗ ∈ Mat p × N ( M ) and C = c ⊗ ∈ Mat N × p ( M ) . Let D ∈ GL N ( M ) and F ∈ Mat p ( M )be the evaluations of d and f , respectively, at X θ = x θ for θ = 1 , . . . , q . Since D is the image of an invertible matrix under a unital algebra homomorphism, D isindeed invertible. Furthermore zI p ⊗ − F is invertible by Lemma 4.3.4. It followsby the discussion after formula (17) that the matrix (cid:20) zI p ⊗ BC D (cid:21) ∈ Mat p + N ( M ((1 /z )))is invertible, and from (17) itself it follows that (cid:20) zI p ⊗ BC D (cid:21) − = (cid:20) D − (cid:21) + (cid:20) I p ⊗ − D − C (cid:21) ( zI p ⊗ − F ) − (cid:2) I p ⊗ − BD − (cid:3) . In turn, by (29) we have L = (cid:20) zI p ⊗ BC D (cid:21) ♮ , hence L is invertible and moreover (33) holds by Lemma 4.3.4. The proof of Propo-sition 4.4.1 is complete. (cid:3) Solving the generalized Schwinger-Dyson equation
We finish the proof of Proposition 2.5.2 by constructing sufficiently many solu-tions of the generalized Schwinger-Dyson equation.5.1.
Statement of the construction.
Here is our main result in this section.
Proposition 5.1.1.
Fix data of the form (7) . Consider the matrix (34) A = − ⊗ a (0) − q X θ =1 X k ∈ N e [ θ ⋆ k, k ] + ∞ X j =1 κ ( θ ) j +1 e [ k, θ ⋄ j ⋆ k ] ⊗ a ( θ ) . constructed by using these data. Assume that G = A − ∈ M ((1 /z )) exists, and (35) g ( i, j ) = G ( i − , j − for i, j = 1 , . . . , n . (36) Then (6) , (8) , and (9) hold, i.e., the data (7) constitute a solution of the generalizedSchwinger-Dyson equation. We complete the proof below in § Remark.
We can rewrite (34) as(37) A = − ⊗ a (0) − q X θ =1 ˆ λ ( θ ) + ∞ X j =1 κ ( θ ) j +1 λ ( θ ) ⊗ a ( θ ) in terms of the lowering and raising operators considered in Remark 4.2.5, thusmaking the relationship of the definition of A to the setup of [35] more transparent.5.1.3. Completion of the proof of Proposition 2.5.2 with Proposition 5.1.1 granted.
We identify X in Proposition 2.5.2 with X in Proposition 4.4.1. We complete thechoice of positive integer q and the given family nn κ ( θ ) j o qθ =1 o ∞ j =2 to a family(38) (cid:18) q, n, g, a (0) , n a ( θ ) o qθ =1 , (cid:26)n κ ( θ ) j o ∞ j =2 (cid:27) qθ =1 (cid:19) of the form (7) where n = p + N > p,a (0) = − L + (cid:20) zI p
00 0 (cid:21) − q X θ =1 L θ ! ∈ Mat n ( C [ z ]) ,a ( θ ) = − L ( θ ) ∈ Mat n ( C ) for θ = 1 , . . . , q and g ( i, j ) = − L − ( i − , j −
1) for i, j = 1 , . . . , n . RESERVATION OF ALGEBRAICITY IN FREE PROBABILITY 23
For the family (38) the hypotheses (35) and (36) of Proposition 5.1.1 are fulfilledby (32) and the definition of g , respectively. Thus (38) is a solution of the gener-alized Schwinger-Dyson equation, i.e., (6), (8), and (9) hold. Property (10) holdsby construction and property (11) holds by (33). The proof of Proposition 2.5.2 iscomplete modulo the proof of Proposition 5.1.1. (cid:3) Lemma 5.1.4.
To prove Proposition 5.1.1 it is necessary only to verify statements (6) and (8) for data (7) satisfying hypotheses (35) and (36) . In the proof below we are reusing elements of the “secondary trick” used in [1]to obtain certain correction terms.
Proof.
The weakened version of Proposition 5.1.1 delivering only conclusions (6)and (8) for data (7) satisfying (35) and (36) we will call Proposition 5.1.1 − ǫ .Our task is to derive Proposition 5.1.1 from Proposition 5.1.1 − ǫ . To that end fix b ∈ Mat n ( C ((1 /z )) arbitrarily and consider new data consisting of(39) a positive integer ˆ n = 2 n (but q the same as before),a matrix ˆ a (0) = (cid:20) a (0) b a (0) (cid:21) ∈ Mat ˆ n ( C ((1 /z ))),matrices ˆ a ( θ ) = (cid:20) a ( θ ) a ( θ ) (cid:21) ∈ Mat ˆ n ( C ) for θ = 1 , . . . , q ,a matrix ˆ g = (cid:20) g h g (cid:21) ∈ Mat ˆ n ( C ((1 /z ))) ( h to be determined), anda family {{ κ ( θ ) j } ∞ j =2 } qθ =1 of complex numbers (same as before).We will apply Proposition 5.1.1 − ǫ to the new data (39) thereby deriving (9) forthe old data (7). To apply Proposition 5.1.1 − ǫ we need first to verify invertibilityof the matrixˆ A = − ˆ a (0) − q X θ =1 X k ∈ N e [ θ ⋆ k, k ] + ∞ X j =1 κ ( θ ) j +1 e [ k, θ ⋄ j ⋆ k ] ⊗ ˆ a ( θ ) . Because ♮ is an isomorphism, there exists unique ˜ A ∈ Mat n ( M ((1 /z ))) such that( ˜ A ) ♮ = A . Using (29) we obtain the relationˆ A = (cid:20) ˜ A − b ⊗ A (cid:21) ♮ ∈ M ((1 /z )) , and we have explicitlyˆ G = (cid:20) ˜ A − b ⊗ A (cid:21) − ! ♮ = (cid:20) ˜ A − ˜ A − ( b ⊗ ) ˜ A − A − (cid:21) ♮ . Thus the new data (39) satisfy (35), and moreover there is a choice of h we canin principle read off from the last displayed line above so that hypothesis (36) issatisfied. (An explicit formula for h is not needed.) Statement (8) of Proposition5.1.1 − ǫ applied to the new data asserts thatlim j →∞ val (cid:20) a ( θ ) g a ( θ ) h a ( θ ) g (cid:21) j = −∞ for θ = 1 , . . . , q . This can also be deduced directly from (8) as it pertains to the old data (7). Finally,the key point is that by statement (6) as it pertains to the new data (39) we have I ˆ n + (cid:20) a (0) b a (0) (cid:21) (cid:20) g h g (cid:21) + q X θ =1 ∞ X j =2 κ ( θ ) j (cid:18)(cid:20) a ( θ ) a ( θ ) (cid:21) (cid:20) g h g (cid:21)(cid:19) j = 0 . Looking in the upper left corners, we obtain an identity a (0) h + bg + q X θ =1 ∞ X j =2 j − X ν =0 κ ( θ ) j ( a ( θ ) g ) ν ( a (0) h )( a ( θ ) g ) j − − ν = 0 . The latter equation, because b is arbitrary and g is invertible by (6) as it pertains tothe old data (7), proves that (9) holds for the old data. In other words, Proposition5.1.1 − ǫ does indeed imply Proposition 5.1.1. (cid:3) Proof of Proposition 5.1.1.
In broad outline the proof is similar to theproof we previously gave for Lemma 3.2.2. But more machinery is needed.5.2.1.
Block decompositions.
Throughout the proof it will be convenient to workwith the block decompositions defined by the formulas A = X i,j ∈ N e [ i, j ] ⊗ A h i, j i and G = X i,j ∈ N e [ i, j ] ⊗ G h i, j i where A h i, j i , G h i, j i ∈ Mat n ( C ((1 /z ))) and in particular G h , i = g. The key point is contained in the following result which says that the first columnof blocks in G has a relatively simple structure. Lemma 5.2.2.
We have (40) G h θ ⋆ · · · ⋆ θ k , i = ga ( θ ) ga ( θ ) · · · ga ( θ k ) g for k ∈ N and θ , . . . , θ k ∈ { , . . . , q } . This relation generalizes formula (20) above.
Proof.
For the proof we will use Lemma 3.3.2 in the case A = M ((1 /z )) , A ∈ A : as on line (34) ,π = e [0 , ⊗ I n ∈ A and σ = ∞ X i =1 e [ i, i ] ⊗ I n ∈ A . We will also use the matrices R ( θ ) = X k ∈ N e [ k ⋆ θ, k ] ⊗ I n ∈ M and ˆ R ( θ ) = X k ∈ N e [ k, k ⋆ θ ] ⊗ I n ∈ M for θ = 1 , . . . , q .These matrices satisfyˆ R ( θ ) R ( θ ′ ) = δ θθ ′ for θ, θ ′ = 1 , . . . , q ,(41) ˆ R ( θ ) AR ( θ ′ ) = δ θθ ′ A for θ, θ ′ = 1 , . . . , q , and(42) q X θ =1 R ( θ ) ˆ R ( θ ) = σ, (43) RESERVATION OF ALGEBRAICITY IN FREE PROBABILITY 25 as can either be verified directly by straightforward if tedious calculation or byusing the definitions and relations (24), (25), (26), (27), and (28) listed above inRemark 4.2.5, along with the rewrite (37) of the definition of A . It follows by (42)and (43) that σAσ = q X θ =1 R ( θ ) ˆ R ( θ ) ! A q X θ =1 R ( θ ′ ) ˆ R ( θ ′ ) ! = q X θ =1 R ( θ ) A ˆ R ( θ ) . It follows in turn that A − σ exists and more precisely that(44) A − σ = q X θ =1 R ( θ ) G ˆ R ( θ ) , as one verifies using (41). We furthermore have σAπA − π = − q X θ =1 ( e [ θ, ⊗ a ( θ ) )( e [0 , ⊗ g ) = − q X θ =1 R ( θ ) ( e [0 , ⊗ a ( θ ) g ) , as one can immediately check. Finally, we have the following chain of equalities: q X θ =1 X i ∈ N e [ i ⋆ θ, ⊗ G h i ⋆ θ, i = σA − π = − A − σ AπA − π = q X θ =1 q X θ ′ =1 R ( θ ) G ˆ R ( θ ) R ( θ ′ ) ( e [0 , ⊗ a ( θ ′ ) g )= q X θ =1 X i ∈ N e [ i ⋆ θ, ⊗ G h i, i a ( θ ) g. At the first step we used Lemma 4.1.5, at the second equation (18) of Lemma 3.3.2,at the third (44), and at the last (41). Thus (40) holds. (cid:3)
Proof of (8) . By definition of M ((1 /z )) we havelim i →∞ val G h i, i = −∞ . Thus by Lemma 5.2.2 we have for θ = 1 , . . . , q thatlim i →∞ val ( a ( θ ) g ) i = lim i →∞ val a ( θ ) G h θ ⋄ i , i = 0 , which proves statement (8).5.2.4. Proof of (6) . Consider the following calculation: I n = X j ∈ N A h , j i G h j, i = − a (0) G h , i − q X θ =1 ∞ X j =1 κ ( θ ) j +1 a ( θ ) G h θ ⋄ j , i = − a (0) g − q X θ =1 ∞ X j =1 κ ( θ ) j +1 ( a ( θ ) g ) j . The first step holds by definition of G , the second by definition of A , and the lastby Lemma 5.2.2. This calculation proves statement (6).5.2.5. Completion of the proof.
By Lemma 5.1.4 it is necessarily the case that state-ment (9) holds. The proof of Proposition 5.1.1 is complete and in turn the proof ofProposition 2.5.2 is complete. (cid:3)
Miscellaneous remarks. GR ( θ ) = ( R ( θ ) − GπAR ( θ ) ) G for θ = 1 , . . . , q .Since statement (45) is not needed for the proof of Theorem 1, we omit its proof.It is easy to see that Lemma 5.2.2 and (45) together allow one to make everyblock G h i, j i explicit in terms of g and A . Doing so in a systematic way one wouldobtain a generalization of Theorem 1 having the full statement of Theorem 2 as aconsequence. 6. Notes on Newton-Puiseux series
At this point in the paper we switch from the noncommutative viewpoint pre-viously stressed to the viewpoint of commutative algebra and algebraic geometry.The latter is maintained throughout the rest of the paper.6.1.
Newton-Puiseux series and Newton polygons.
We review basic devicesfor understanding singularities of plane algebraic curves in characteristic zero andmake some definitions needed for later calculations.6.1.1.
The algebraic closure of C ((1 /z )) . Let K denote the union of the tower offields { C ((1 /z /n ! )) } ∞ n =1 . In other words, K arises by adjoining roots of z of all ordersto C ((1 /z )). We call an element of K a Newton-Puiseux series . When discussing K below we often use the more apposite abbreviated notation K = C ((1 /z )). Itis well-known that K is the algebraic closure of K . See, e.g., [14, Cor. 13.15]. Theoriginal insight is due to Newton.6.1.2. Extension of the valuation function val to K . Each element f ∈ K has bydefinition a unique Newton-Puiseux expansion f = P u ∈ Q c u z u with coefficients c u ∈ C such that for some positive integer N depending on f one has c u = 0 unless u ≤ N and N u ∈ Z . To extend to K the valuation defined on K , we defineval f = sup { u ∈ Q | c u = 0 } ∈ Q ∪ {−∞} . The properties (2), (3), and (4) continue to hold for the extension of val to K . RESERVATION OF ALGEBRAICITY IN FREE PROBABILITY 27
Proposition 6.1.3.
Let P ( y ) ∈ K [ y ] be a polynomial of degree n in a variable y with coefficients in the field K . Write P ( y ) = n X i =0 a i y n − i = a n Y i =1 ( y − r i ) ( a i ∈ K and r i ∈ K ) , enumerating the roots r i so that val r ≥ · · · ≥ val r n . (i) Then we have val a i ≤ val a + val r · · · r i for i = 0 , . . . , n , (46) with equality for i = 0 , n and furthermore (47) for i = 1 , . . . , n − s.t. val r i > val r i +1 .(ii) Let ψ : [0 , n ] → R ∪ {−∞} be the infimum of all affine linear functions λ : [0 , n ] → R satisfying λ ( i ) ≥ val a i for i = 0 , . . . , n . Then we have the inte-gral formula ψ ( s ) = val a + R s val r ⌈ u ⌉ du . Here ⌈ u ⌉ denotes the least integer not less than u . The function ψ is concaveby construction. The function ψ (or rather, its graph) is the Newton polygon as-sociated with P ( y ), up to reflections in and translations parallel to the horizon-tal and vertical axes (conventions vary from author to author). Similarly, a New-ton polygon is attached to any one-variable polynomial with coefficients in a dis-cretely valued field. For background on Newton polygons see [6, Chap. 2, Sec. 5] or[9, Part III, Chap. 8, Sec. 3]. Proof. (i) Since ( − i a i /a for i > i th symmetric function of the roots r , . . . , r n , the result follows straightforwardly from (2), (3) and (4). (ii) The de-duction of this statement from the preceding one is standard. (cid:3) Extension of the valuation to
Mat n ( K ) . We extend val from Mat n ( K ) toMat n ( K ) by the rule val A = max ni,j =1 val A ( i, j ). Then Mat n ( K ) satisfies all theaxioms of a Banach algebra over K except completeness. Lack of completeness willnot be an issue.6.2. Applications.
We present several applications of the preceding machineryneeded for the proof of Proposition 2.5.3.6.2.1.
Specialized matrix notation.
Given A ∈ Mat n ( C ((1 /z ))), we define e ( A ) = ( e ( A ) , . . . , e n ( A )) ∈ C ((1 /z )) n by the formuladet( tI n − A ) = t n + n X i =1 e i ( A )( − i t n − i ∈ C ((1 /z ))[ t ] . The Cayley-Hamilton Theorem takes then the form(48) A n + n X i =1 ( − i e i ( A ) A n − i = 0 . Proposition 6.2.2.
For A ∈ Mat n ( C ((1 /z ))) the following are equivalent: (I) e ( A ) ∈ (1 /z ) C [[1 /z ]] n . (II) lim k →∞ val A k = −∞ .Proof. We add one further statement to the list above:(III) Every eigenvalue of A in K has (strictly) negative valuation. Statements (I) and (III) are equivalent by Proposition 6.1.3. It remains only toprove the equivalence (II) ⇔ (III). It is actually easier to prove more. We will provethe equivalence (II) ⇔ (III) for A ∈ Mat n ( K ). Supposing at first that A consists ofa single Jordan block, one verifies the equivalence by inspection. In general we canwrite A = W JW − where W ∈ GL n ( K ) and J ∈ Mat n ( K ) is block-diagonal withdiagonal blocks of the Jordan form and we have a bound (cid:12)(cid:12) val A k − val J k (cid:12)(cid:12) ≤ val W − + val W which establishes the equivalence (II) ⇔ (III) in general. (cid:3) Negative spectral valuation.
We say that A ∈ Mat n ( K ) has negative spectralvaluation if the equivalent conditions (I) and (II) above hold.6.2.4. Algebraic and nonsingular algebraic elements of C [[ t ]] . In some situations1 /z rather than z is the natural parameter to work with. We therefore make thefollowing definitions which are in principle redundant but in practice convenient. Wesay that f ( t ) ∈ C [[ t ]] is algebraic if F ( t, f ( t )) = 0 for some 0 = F ( x, y ) ∈ C [ x, y ].Of course f ( t ) ∈ C [[ t ]] is algebraic if and only if f (1 /z ) ∈ C ((1 /z )) is algebraicin the sense defined in § f ( t ) nonsingular algebraic if there exists F ( x, y ) ∈ C [ x, y ] satisfying ∂F∂y (0 , f (0)) = 0 and F ( t, f ( t )) = 0.The next statement is the key to desingularization. Lemma 6.2.5.
Let f ( t ) ∈ t C [[ t ]] be a power series with vanishing constant term.Let F ( x, y ) ∈ C [ x, y ] be a polynomial not divisible by x such that F ( t, f ( t )) = 0 .Then the following conditions are equivalent: (I) F (1 /z, y ) ∈ K [ y ] has exactly one root in K of negative valuation. (II) ∂F∂y (0 , = 0 .Proof. Write F ( x, y ) = P ni =0 p i ( x ) y n − i where n > p i ( x ) ∈ C [ x ] and p ( x ) = 0.For p ( x ) ∈ C [ x ], let ord p ( x ) denote the exponent of the highest power of x dividing p ( x ). Let φ : [0 , n ] → R ∪ { + ∞} denote the supremum of all affine linear functions λ : [0 , n ] → R such that λ ( i ) ≤ ord p i ( x ) for i = 0 , . . . , n . The function φ is convex.Since F (0 ,
0) = 0 we have φ ( n ) >
0. Since x does not divide F ( x, y ), we havemin ni =0 ord p i ( x ) = min ni =0 φ ( i ) = 0. Let i be the maximum of i = 0 , . . . , n − φ ( i ) = 0. Statement (ii) is equivalent to the assertion that i = n − F (1 /z, y ) has n − i roots of negative valuation by Proposition 6.1.3 and theobservation that ord p ( x ) = − val p (1 /z ). Thus statement (I) is equivalent to theassertion that i = n − (cid:3) The next statement summarizes just enough of the theory of resolution of sin-gularities of plane algebraic curves in characteristic zero for our purposes.
Proposition 6.2.6.
Let P ∞ i =0 c i t i ∈ t C [[ t ]] ( c i ∈ C ) be an algebraic power series.Then P ∞ i = N c i + N t i ∈ t N C [[ t ]] is a nonsingular algebraic power series for all N ≫ .Proof. Let f = P ∞ i =1 c i z − i ∈ C ((1 /z )) and f N = P ∞ i = N c i + N z − i ∈ C ((1 /z )). Let F ( x, y ) ∈ C [ x, y ] (resp., F N ( x, y ) ∈ C [ x, y ]) denote the irreducible equation (see § f (resp., f N ). It will be enough to show that F N (1 /z, y ) ∈ K [ y ] hasexactly one root in K of negative valuation for N ≫
0. If f N = 0 for some N then f N = 0 for all N ≥ N and there is nothing to prove. Thus we may as-sume without loss of generality that f N = 0 for all N ≥
0. Let n denote thedimension of C ( z, f ) over C ( z ). It is clear that f and f N generate the same ex-tension of C ( z ). Thus F ( x, y ) and F N ( x, y ) have the same degree in y , namely RESERVATION OF ALGEBRAICITY IN FREE PROBABILITY 29 n . Let r , . . . , r n denote the roots in K of the polynomial F (1 /z, y ) ∈ K [ y ],enumerated so that r n = f . Let h N = P N − i =0 a i z − i = f − f N /z N ∈ C ( z ),in which case necessarily f N = z N ( f − h N ). Then for a suitable enumeration r ,N , . . . , r n,N of the roots in K of the polynomial F N (1 /z, y ) ∈ K [ y ], we have r i,N = z N ( r i − h N ) for i = 1 , . . . , n and r n,N = f N . Now the roots r , . . . , r n aredistinct due to irreducibility of F ( x, y ), and clearly val ( f − h N ) ≤ − N . Because h N → N →∞ f with respect to the valuation val, it follows that for some integer N > f , and all integers N ≥ N , we have n − min i =1 val r i,N = N + n − min i =1 val( r i − f ) ≥ > − N ≥ val f N = val r n,N , whence the result via Lemma 6.2.5. (cid:3) Evaluation of algebraic power series on matrices
The main result of this section is Proposition 7.3.4 below. The main tools usedin this section are the Cayley-Hamilton Theorem, the Weierstrass Preparation The-orem, and Proposition 6.2.6 above.7.1.
Motivation.
Let A be an n -by- n matrix with complex entries. We take ourinspiration from the undergraduate level approach in [39] to computing the matrixexponential exp( tA ). The approach is lengthy as presented for sophomores but itcan be summarized quickly at graduate level as follows. Perform Weierstrass division(possible globally in this case) in order to obtain an identity relating two-variableentire functions of complex variables t and X , namely(49) exp( tX ) = n − X k =0 y k ( t ) X k + Q ( X, t ) det( XI n − A ) , for suitable and unique remainder P n − k =0 y k ( t ) X k and quotient Q ( X, t ). Differenti-ation of (49) on both sides with respect to t yields a first order homogeneous lineardifferential equation for the vector function y ( t ) = ( y ( t ) , . . . , y k ( t )) which togetherwith the evident initial value data uniquely determines y ( t ). One can then go onto solve explicitly for the functions y k ( t ) in closed form. By plugging in X = A onboth sides of (49) and using the Cayley-Hamilton Theorem one then has finally(50) exp( tA ) = n − X k =0 y k ( t ) A k . Formula (50) makes no reference to the Jordan canonical form of A . Indeed, byconstruction, the coefficients y k ( t ) are uniquely determined by the characteristicpolynomial of A alone. We will make roughly analogous use of Weierstrass divisionbelow to evaluate algebraic power series on matrices with entries in C ((1 /z )) ofnegative spectral valuation.7.2. I -adic convergence, power series, and Weierstrass division. We pauseto review generalities connected with the formal power series version of the Weier-strass Preparation Theorem. I -adic convergence. Given a commutative ring R with unit, an ideal I , and asequence { a }∪{ a i } ∞ i =1 in R , one says lim i →∞ a i = a holds I -adically if for every posi-tive integer k there exists a positive integer i = i ( k ) such that a − a i ∈ I k for all i ≥ i . Similarly, one can speak of I -adic Cauchy sequencesand I -adic completeness. Consider, e.g., the ring C [[ u , . . . , u n ]] = C [[ u ]] and themaximal ideal I = ( u , . . . , u n ) ⊂ C [[ u ]]. Then f i ∈ C [[ u ]] converges I -adicallyto f ∈ C [[ u ]] if and only if for every n -tuple ( ν , . . . , ν n ) of nonnegative integersand every sufficiently large index i depending on ( ν , . . . , ν n ), the Taylor coefficient ν ! ··· ν n ! ∂ ν ··· + νn f i ∂u ν ··· ∂u νnn (0) equals the Taylor coefficient ν ! ··· ν n ! ∂ ν ··· + νn f∂u ν ··· ∂u νnn (0). It is easyto see that the ring C [[ u ]] is I -adically complete.7.2.2. Weierstrass division.
We now briefly recall the
Weierstrass Preparation The-orem from a more active point of view emphasizing the algorithm of Weierstrassdivision. See, e.g., [42, Thm. 5, p. 139, Chap. VII, §
1] for background and proof. Thetheorem concerns an ( n + 1)-variable power series ring over a field, with one of thevariables singled out for special treatment. For definiteness we take the coefficientfield to be C . Consider the ring C [[ u , . . . , u n , t ]] = C [[ u, t ]], with t distinguished.One is given a divisand F ( u, t ) ∈ C [[ u, t ]] and a divisor D ( u, t ) ∈ C [[ u, t ]]. Of thelatter it is assumed that there exists a positive integer m (called the multiplicity ofthe divisor) such that D (0 , t ) = t m U ( t ) for some U ( t ) ∈ C [[ t ]] such that U (0) = 0.The Weierstrass division process delivers a quotient Q ( u, t ) ∈ C [[ u, t ]] and a re-mainder R ( u, t ) ∈ C [[ u ]][ t ]. The pair ( Q ( u, t ) , R ( u, t )) is uniquely determined bytwo requirements. Firstly, the division equation F ( u, t ) = Q ( u, t ) D ( u, t ) + R ( u, t )must hold. Secondly, R ( u, t ) must be a polynomial in t of degree < m . It bears em-phasis that if D ( u, t ) ∈ C [[ u ]][ t ] is monic of degree m such that D (0 , t ) = t m , and F ( u, t ) ∈ C [[ u ]][ t ], then the Euclidean (i.e., high school) and Weierstrass divisionprocesses deliver the same quotient and remainder. Lemma 7.2.3.
We continue in the setting of the preceding paragraph. However, forsimplicity we assume now that D ( u, t ) is monic of degree m such that D (0 , t ) = t m . Consider the ideal I = ( u , . . . , u n ) ⊂ C [[ u ]][ t ] . Let k be a posi-tive integer. If t k divides F ( u, t ) , then R ( u, t ) belongs to the ideal I ⌊ k/m ⌋ . (Here ⌊ c ⌋ denotes the greatest integer not exceeding c .) It follows that formation of Weierstrass remainder upon division by D ( u, t )viewed as a function from C [[ u, t ]] to C [[ u ]][ t ] is continuous with respect to the( t )-adic topology on the source and the ( u , . . . , u n )-adic topology on the target. Proof.
Let F ( u, t ) = F ( u, t ) /t k . Let R ( u, t ) denote the remainder of F ( u, t )upon Weierstrass division by D ( u, t ). Then R ( u, t ) is the remainder of t k R ( u, t )upon high school division by D ( u, t ). This noted, there is no loss of generalityin assuming that F ( u, t ) = t k . Write D ( u, t ) = t m + P m − i =0 a i t i with coefficients a i = a i ( u ) ∈ C [[ u ]] such that a i (0) = 0. Write R ( u, t ) = P m − i =0 b i t i with coefficients b i = b i ( u ) ∈ C [[ u ]]. Then we have − a − a m − k = b ... b m − , RESERVATION OF ALGEBRAICITY IN FREE PROBABILITY 31 where the matrix on the left is the so-called companion matrix for D ( u, t ). Clearlyevery entry of the m th power of the companion matrix belongs to the ideal I , andhence every entry of the k th power belongs to the ideal I ⌊ k/m ⌋ . (cid:3) Formulation of the main result.
We state a technical result needed tomake the final arguments of the proof of Proposition 2.5.3.7.3.1.
Variables and rings.
Throughout the remainder of § n and we work with the family of independent (commuting) algebraic variables { u i } ni =1 ∪ { v i } ni =1 ∪ { t, x, y } . Let u = ( u , . . . , u n ) and v = ( v , . . . , v n ) . We write C [ u ] = C [ u , . . . , u n ] , C [[ u ]] = C [[ u , . . . , u n ]] , C [ u, v ] = C [ u , . . . , u n , v , . . . , v n ] , and so on. We use similar notation below for building up rings from the given vari-ables without further comment. Given, for example P ( u, v ) ∈ C [ u, v ] n , we denoteby ∂P∂u ( u, v ) the 2 n -by- n matrix with entries ∂P i ∂u j ( u, v ). We use similar notation forderivatives of vector functions below without further comment.7.3.2. Specialized matrix notation.
Given A ∈ Mat n ( C ((1 /z ))), let A ♭ = (cid:2) A (1 , . . . A ( n, . . . A (1 , n ) . . . A ( n, n ) (cid:3) T ∈ Mat n × ( C ((1 /z ))) . Note that(51) ( BA ) ♭ = ( I n ⊗ B ) A ♭ for B ∈ Mat n ( C ((1 /z ))).7.3.3. Setup for the main result.
We are given an algebraic power series(52) f ( t ) = ∞ X i =0 c i t i ∈ C [[ t ]] ( c i ∈ C )and a matrix A ∈ Mat n ( C ((1 /z )))of negative spectral valuation, i.e., a matrix satisfying the conditionslim k →∞ val A k = −∞ and e ( A ) ∈ (1 /z ) C [[1 /z ]] n which by Proposition 6.2.2 are equivalent. Proposition 7.3.4.
Notation and assumptions are as above. For every N ≥ suchthat P ∞ i = N c i + N t i ∈ C [[ t ]] is nonsingular algebraic and divisible by t n , there exist γ ∈ (1 /z ) C [[1 /z ]] n and P ( u, v ) ∈ C [ u, v ] n such that the following statements hold: P ( e ( A ) , γ ) = 0 . (53) det ∂P∂v ( e ( A ) , γ ) = 0 . (54) (cid:2) ( A ) ♭ . . . ( A n − ) ♭ (cid:3) (cid:18) ∂P∂v ( e ( A ) , γ ) (cid:19) − ∂P∂u ( e ( A ) , γ ) = 0 . (55) ∞ X i = N c i + N (cid:20) A ⊗ I n I n ⊗ I n I n ⊗ A (cid:21) i = n X i =1 γ i (cid:20) A ⊗ I n I n ⊗ I n I n ⊗ A (cid:21) i − . (56)The proof takes up the rest of § § N ≥ n sufficiently large depending on f ( t ) satisfies the hypotheses of Proposition 7.3.4 byProposition 6.2.6.7.3.5. Remark.
In the application to the proof of Proposition 2.5.3 we will need touse several consequences of the conclusions of Proposition 7.3.4 which are easy tocheck once written down but might otherwise be obscure. For the reader’s conve-nience we write these consequences down. Firstly, we observe that the statement (cid:2) ( A N ) ♭ . . . ( A N +2 n − ) ♭ (cid:3) (cid:18) ∂P∂v ( e ( A ) , γ ) (cid:19) − ∂P∂u ( e ( A ) , γ ) = 0(57)follows from statement (55) via statement (51). Secondly, we observe that (56)implies ∞ X i = N c i + N (cid:20) A B A (cid:21) i = n X i =1 γ i (cid:20) A B A (cid:21) i − for B ∈ Mat n ( C ((1 /z )))(58)by (so to speak) substituting B for ⊗ . Finally, we observe that ∞ X i =2 N c i A i = n X i =1 γ i A N + i − and(59) ∞ X i =2 N i − X ν =0 c i A ν BA i − − ν = n X i =1 N + i − X ν =0 γ i A ν BA i − − ν for B ∈ Mat n ( C ((1 /z ))).(60)These last two statements are obtained by right-multiplying statement (58) on bothsides by (cid:20) A B A (cid:21) N and expanding the matrix powers.7.3.6. Reduction of the proof of Proposition 7.3.4.
After replacing f ( t ) by P ∞ i = N c i + N t i , we may assume without loss of generality that N = 0. Thus weare making two further special assumptions concerning f ( t ) which for the sake ofclarity and convenient reference we write out explicitly. Firstly, we are assumingthat(61) f ( t ) ∈ t n C [[ t ]] . Secondly, we are assuming that there exists F ( x, y ) ∈ C [ x, y ] RESERVATION OF ALGEBRAICITY IN FREE PROBABILITY 33 such that(62) F ( t, f ( t )) = 0 and ∂F∂y (0 , = 0 . Note that the formula(63) F (0 ,
0) = ∂F∂x (0 ,
0) = · · · = ∂ n − F∂x n − (0 ,
0) = 0follows straightforwardly from (61) and (62).7.4.
A candidate for γ . A special polynomial.
Let(64) D ( u, t ) = t n + n X i =1 ( − i u i t n − i ∈ C [ u, t ] . Note that the left side of (48) equals D ( e ( A ) , A ). This is the motivation for thedefinition of D ( u, t ).7.4.2. Construction of ϕ ( u ) . Perform Weierstrass division by D ( u, t ) to obtain anidentity(65) f ( t ) = n X i =1 ϕ i ( u ) t i − + Q ( u, t ) D ( u, t ) , where ϕ ( u ) = ( ϕ ( u ) , . . . , ϕ n ( u )) ∈ u C [[ u ]] n and Q ( u, t ) ∈ C [[ u, t ]] . Note that we indeed have ϕ (0) = 0 as one verifies by substituting u = 0 on bothsides of (65) and using hypothesis (61). Lemma 7.4.3.
We have ∞ X j =0 c j (cid:20) A ⊗ I n I n ⊗ I n I n ⊗ A (cid:21) j = n X j =1 ϕ j ( e ( A )) (cid:20) A ⊗ I n I n ⊗ I n I n ⊗ A (cid:21) j − and (66) (cid:2) ( A ) ♭ . . . ( A n − ) ♭ (cid:3) ∂ϕ∂u ( e ( A )) = 0 . (67)Thus the reasonable candidate for γ is ϕ ( e ( A )). Proof.
Note that (67) can be rewritten(68) n X i =1 ∂ϕ i ∂u j ( e ( A )) A i − = 0 for j = 1 , . . . , n .Note also that by differentiation we deduce from (65) that n X i =1 ∂ϕ i ∂u j ( u ) t i − (69) = − (cid:18) ∂Q ∂u j ( u, t ) D ( u, t ) + 2 Q ( u, t ) ∂D∂u j ( u, t ) (cid:19) D ( u, t ) for j = 1 , . . . , n . Now suppose temporarily that f ( t ) ∈ C [ t ]. Then we have ϕ ( u ) ∈ u C [ u ] n and Q ( u, t ) ∈ C [ u, t ] since high school division in this case gives the same result asWeierstrass division. Substituting( u, t ) = (cid:18) e ( A ) , (cid:20) A ⊗ I n I n ⊗ I n I n ⊗ A (cid:21)(cid:19) into (65) and using the Cayley-Hamilton Theorem (48), we obtain (66). Substituting( u, t ) = ( e ( A ) , A ) into (69) and using the Cayley-Hamilton Theorem (48) again, weobtain (68). The general case follows by a routine approximation argument basedon Lemma 7.2.3 and the remark immediately following. (cid:3) A candidate for P ( u, v ) . Construction of the candidate.
Perform high school division of F t, n X i =1 v i t i − ! ∈ C [ u, v, t ]by D ( u, t ) to obtain an identity(70) F t, n X i =1 v i t i − ! = n X i =1 P i ( u, v ) t i − + Q ( u, v, t ) D ( u, t ) where Q ( u, v, t ) ∈ C [ u, v, t ]and P ( u, v ) = ( P ( u, v ) , . . . , P n ( u, v )) ∈ C [ u, v ] n . The latter is our candidate for P ( u, v ). Lemma 7.5.2.
Assumptions and notation are as above. Then the following state-ments hold: P ( u, ϕ ( u )) = 0 . (71) det ∂P∂v (0 , = 0 and hence ∂P∂v ( u, ϕ ( u )) ∈ GL n ( C [[ u ]]) . (72) ∂ϕ∂u ( u ) = (cid:18) ∂P∂v ( u, ϕ ( u )) (cid:19) − ∂P∂u ( u, ϕ ( u )) . (73)7.5.3. Proof of (71) . Perform Weierstrass division of F ( x, y ) by y − f ( x ) in theformal power series ring C [[ x, y ]] to obtain the identity(74) F ( x, y ) = ( y − f ( x )) U ( x, y )for some U ( x, y ) ∈ C [[ x, y ]]. A priori one should add a remainder term r ( x ) ∈ C [[ x ]]to the right side but substitution of y = f ( x ) on both sides and the hypothesis (62) RESERVATION OF ALGEBRAICITY IN FREE PROBABILITY 35 shows that r ( x ) = 0. We then have the following chain of equalities: n X i =1 P i ( u, ϕ ( u )) t i − + Q ( u, ϕ ( u ) , t ) D ( u, t ) = F t, n X i =1 ϕ ( u ) t i − ! = n X i =1 ϕ ( u ) t i − − f ( t ) ! U t, n X i =1 ϕ i ( u ) t i − ! = − Q ( u, t ) D ( u, t ) U t, n X i =1 ϕ ( u ) t i − ! . Justifications for the steps are as follows. The first equality we obtain by substi-tuting v = ϕ ( u ) into (70). The second equality we obtain by substituting into thefactorization given in (74). The third equality we obtain by rearrangement of (65).The equality between the extreme terms of the chain of equalities above forces (71)to hold by the uniqueness of the remainder produced by Weierstrass division.7.5.4. Proof of (72) . Differentiation on both sides of (70) with respect to v j followedby evaluation at u = v = 0 yields the relation(75) ∂F∂y ( t, t j − = n X i =1 ∂P i ∂v j (0 , t i − + ∂Q ∂v j (0 , , t ) t n . Now write ∂F∂y ( x,
0) = ∞ X i =0 b i x i ( b i ∈ C ) . By (75) we have ∂P i ∂v j (0 ,
0) = (cid:26) b i − j if j ≤ i ,0 if j > i for i, j = 1 , . . . , n . Thus we have n det i,j =1 ∂P i ∂v j (0 ,
0) = (cid:18) ∂F∂y (0 , (cid:19) n . The right side does not vanish by assumption (62). Thus (72) holds.7.5.5.
Proof of (73) . Formula (73) holds by implicit differentiation of formula (71).The proof of Lemma 7.5.2 is complete. (cid:3)
Completion of the proof of Proposition 7.3.4.
As noted above, we mayassume that N = 0 and hence that assumptions (61), (62) and (63) are in force.Property (53) follows from formula (71). Property (54) follows from formula (72).Property (55) follows from formulas (67) and (73). Property (56) follows from for-mula (66). The proof of Proposition 7.3.4 is complete. (cid:3) Proof of the main result
We finish the proof of Proposition 2.5.3 by checking hypotheses in Proposition2.4.2, thereby completing the proof of Theorem 1.
Review of the setup for Proposition 2.5.3.
Let us start simply by re-peating statements (6) and (9) here for the reader’s convenience:(76) I n + a (0) g + q X θ =1 ∞ X j =2 κ ( θ ) j ( a ( θ ) g ) j = 0 . The linear map(77) h a (0) h + q X θ =1 ∞ X j =2 j − X ν =0 κ ( θ ) j ( a ( θ ) g ) ν ( a ( θ ) h )( a ( θ ) g ) j − − ν : Mat n ( C ((1 /z ))) → Mat n ( C ((1 /z ))) is invertible.Concerning the data appearing above, we have by (7) and (12) that(78) a (0) ∈ Mat n ( C ( z )) , a (1) , . . . , a ( q ) ∈ Mat n ( C ) and g ∈ Mat n ( C ((1 /z ))) . Application of Proposition 7.3.4.
By (7) and (13) we have that P ∞ j =2 κ ( θ ) j t j ∈ C [[ t ]] is algebraic for θ = 1 , . . . , q .By assumption (8) and Proposition 6.2.2 we have thatlim k →∞ q max θ =1 val ( a ( θ ) g ) k = 0 and e ( a (1) g ) , . . . , e ( a ( q ) g ) ∈ (1 /z ) C [[1 /z ]] n . Thus by Propositions 6.2.6 and 7.3.4 along with the remarks immediately followingthe latter, there exist an integer N ≥ γ (1) , . . . , γ ( q ) ∈ (1 /z ) C [[1 /z ]] n and P (1) ( u, v ) , . . . , P ( q ) ( u, v ) ∈ C [ u, v ] n such that P ( θ ) ( e ( a ( θ ) g ) , γ ( θ ) ) = 0 , (79) det ∂P ( θ ) ∂v ( e ( a ( θ ) g ) , γ ( θ ) ) = 0 , (80) ∞ X j =2 N κ ( θ ) j ( a ( θ ) g ) j = n X j =1 γ ( θ ) j ( a ( θ ) g ) N + j − , (81) (cid:2) (( a ( θ ) g ) N ) ♭ . . . (( a ( θ ) g ) N +2 n − ) ♭ (cid:3) (82) × (cid:18) ∂P ( θ ) ∂v ( e ( a ( θ ) g ) , γ ( θ ) ) (cid:19) − ∂P ( θ ) ∂u ( e ( a ( θ ) g ) , γ ( θ ) ) = 0 , and ∞ X j =2 N j − X ν =0 κ ( θ ) j ( a ( θ ) g ) ν ( a ( θ ) h )( a ( θ ) g ) j − − ν (83) = n X j =1 N + j − X ν =0 γ ( θ ) j ( a ( θ ) g ) ν ( a ( θ ) h )( a ( θ ) g ) N + j − − ν for θ = 1 , . . . , q and any h ∈ Mat n ( C ((1 /z ))).8.3. Polynomial version of (76) . We embed (76) into a system of 3 nq + n polynomial equations in 3 nq + n variables with coefficients in C ( z ). RESERVATION OF ALGEBRAICITY IN FREE PROBABILITY 37
Variables.
We employ the family of variables { U i } qni =1 ∪ { V i } qni =1 ∪ { ξ i } n i =1 . Let U = ( U , . . . , U qn ) , V = ( V , . . . , V qn ) , and ξ = ( ξ , . . . , ξ n ) . Let Ξ = ξ . . . ξ n − n +1 ... ... ξ n . . . ξ n ∈ Mat n ( C [ ξ ]) . We break the U ’s and V ’s down into groups by introducing the following notation: U ( θ ) i = U i +( θ − q for i = 1 , . . . , n and θ = 1 , . . . , q . V ( θ ) i = V i +( θ − q for i = 1 , . . . , n and θ = 1 , . . . , q . U ( θ ) = ( U ( θ )1 , . . . , U ( θ ) n ) and V ( θ ) = ( V ( θ )1 , . . . , V ( θ )2 n ) for θ = 1 , . . . , q .8.3.2. A special matrix.
We define H ( V, ξ )= I n + a (0) Ξ + q X θ =1 N − X k =2 κ ( θ ) k ( a ( θ ) Ξ) k + n X k =1 V ( θ ) k ( a ( θ ) Ξ) N + k − ! ∈ Mat n ( C ( z )[ V, ξ ]) . Polynomials.
We define 3 qn + n polynomials belonging to C ( z )[ U, V, ξ ] asfollows. F i + n ( θ − ( U, ξ )= F ( θ ) i ( U ( θ ) , ξ ) = U ( θ ) i − e i ( a ( θ ) Ξ) for i = 1 , . . . , n and θ = 1 , . . . , q , G i +2 n ( θ − ( U, V )= P ( θ ) ( U ( θ ) , V ( θ ) ) for i = 1 , . . . , n and θ = 1 , . . . , q , and H i + n ( j − ( V, Ξ)= H ( V, Ξ)( i, j ) for i, j = 1 , . . . , n .8.3.4.
Presentation of the system of equations.
Let F ( U, ξ ) = (cid:2) F ( U, ξ ) . . . F qn ( U, ξ ) (cid:3) T ,G ( U, V ) = (cid:2) G ( U, V ) . . . G qn ( U, V ) (cid:3) T ,H ( V, ξ ) = (cid:2) H ( V, ξ ) . . . H n ( V, ξ ) (cid:3) T . Then our system of polynomial equations takes the form(84) F ( U, ξ ) = 0 , G ( U, V ) = 0 , H ( V, ξ ) = 0 . Note that this system has all coefficients in C ( z ) by (78) and the definitions. The solution Υ . We claim that the following formulas specify a solutionover C ((1 /z )) of the system of equations (84): U ( θ ) i = e i ( a ( θ ) g ) for i = 1 , . . . , n and θ = 1 , . . . , q .(85) V ( θ ) i = γ ( θ ) i for i = 1 , . . . , n and θ = 1 , . . . , q .(86) ξ i + n ( j − = Ξ( i, j ) = g ( i, j ) for i, j = 1 , . . . , n .(87)The equation F ( U, ξ ) = 0 is obviously satisfied. The equation G ( U, V ) = 0 issatisfied because it merely restates the system of equations (79). Finally, one verifiesthat H ( U, ξ ) = 0 is satisfied by using (76), (81), (85), and (86). The claim is proved.The solution of (84) specified by (85), (86), and (87) will be denoted by Υ .8.4. Analysis of the Jacobian determinant.
Now we study the Jacobian matrix(88) ∂F∂U ( U, ξ ) 0 ∂F∂ξ ( U, ξ ) ∂G∂U ( U, V ) ∂G∂V ( U, V ) 00 ∂H∂V ( V, ξ ) ∂H∂ξ ( V, ξ ) ∈ Mat n +3 qn ( C ( z )[ U, V, ξ ])for the system of equations (84). Let(89) I n b b b b b ∈ Mat n +3 qn ( C ((1 /z )))be the result of evaluating (88) at the point Υ . To prove Proposition 2.5.3 andthereby to complete the proof of Theorem 1, we have by Proposition 2.4.2 only toprove that the determinant of the matrix (89) does not vanish. Now provided thatdet b = 0, we have a matrix identity I n b b b b b I n − b − b I n
00 0 I n I n − b I n
00 0 I n = I n b − b b − b b b b − b b + b . Thus it will be enough to prove thatdet b = 0 , (90) b b − b = 0 , and(91) det b = 0 . (92)8.5. Proof of (90) . We have by the definitions(93) b = q X θ =1 e θθ ⊗ ∂P ( θ ) ∂v ( e ( a ( θ ) g ) , γ ( θ ) ) . Thus (90) holds by (80).
RESERVATION OF ALGEBRAICITY IN FREE PROBABILITY 39
Proof of (91) . For b and b we have formulas similar to (93), namely b = q X θ =1 e θθ ⊗ ∂P ( θ ) ∂u ( e ( a ( θ ) g ) , γ ( θ ) ) and b = q X θ =1 e θθ ⊗ (cid:2) (( a ( θ ) g ) N ) ♭ . . . (( a ( θ ) g ) N +2 n − ) ♭ (cid:3) . Thus (91) holds by (82).8.7.
Proof of (92) . We have for i, j = 1 , . . . , n that ∂ H ( V, ξ ) ∂ξ i +( j − n = a (0) e ij + q X θ =1 2 N − X k =2 k − X ν =0 κ ( θ ) k ( a ( θ ) Ξ) ν ( a ( θ ) e ij )( a ( θ ) Ξ) k − − ν + q X θ =1 2 n X k =1 N + k − X ν =0 V ( θ ) k ( a ( θ ) Ξ) ν ( a ( θ ) e ij )( a ( θ ) Ξ) N + k − − ν and hence after evaluating both sides at Υ and using (83), we find that b is amatrix describing with respect to the basis e , . . . , e n , . . . , e n , . . . , e nn ∈ Mat n ( C ((1 /z )))the invertible linear map considered in (77). Thus (92) holds. Thus the proof ofProposition 2.5.3 is complete and with it the proof of Theorem 1. Acknowledgements:
I thank Serban Belinschi, J. William Helton, Tobias Mai,and Roland Speicher for communications concerning the self-adjoint linearizationtrick and symmetric realization. I thank Steven Lalley for communications concern-ing algebraicity in relation to random walk. I thank Christine Berkesch Zamaerefor communications concerning algebraicity criteria and commutative algebra.
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