aa r X i v : . [ m a t h . R A ] J a n HILBERT’S 17TH PROBLEM IN FREE SKEW FIELDS
JURIJ VOL ˇCI ˇC ‹ Abstract.
This paper solves the rational noncommutative analog of Hilbert’s 17thproblem: if a noncommutative rational function is positive semidefinite on all tuplesof hermitian matrices in its domain, then it is a sum of hermitian squares of noncom-mutative rational functions. This result is a generalization and culmination of earlierpositivity certificates for noncommutative polynomials or rational functions without her-mitian singularities. More generally, a rational Positivstellensatz for free spectrahedra isgiven: a noncommutative rational function is positive semidefinite or undefined at everymatricial solution of a linear matrix inequality L ľ L . The essential intermediate step towards thisPositivstellensatz for functions with singularities is an extension theorem for invertibleevaluations of linear matrix pencils. Introduction
In his famous problem list of 1900, Hilbert asked whether every positive rationalfunction can be written as a sum of squares of rational functions. The affirmative answerby Artin in 1927 laid ground for the rise of real algebraic geometry [BCR98]. Severalother sum-of-squares certificates (Positivstellens¨atze) for positivity on semialgebraic setsfollowed; since the detection of sums of squares became viable with the emergence of semi-definite programming [WSV12], these certificates play a fundamental role in polynomialoptimization [Las01, BPT13].Positivstellens¨atze are also essential in the study of polynomial and rational inequali-ties in matrix variables, which splits into two directions. The first one deals with inequal-ities where the size of the matrix arguments is fixed [PS76, KˇSV18]. The second directionattempts to answer questions about positivity of noncommutative polynomials and ra-tional functions when matrix arguments of all finite sizes are considered. Such questionsnaturally arise in control systems [dOHMP09], operator algebras [Oza16] and quantuminformation theory [DLTW08, P-KRR+19]. This (dimension-)free real algebraic geome-try started with the seminal work of Helton [Hel02] and McCullough [McC01], who provedthat a noncommutative polynomial is positive semidefinite on all tuples of hermitian ma-trices precisely when it is a sum of hermitian squares of noncommutative polynomials.The purpose of this paper is to extend this result to noncommutative rational functions.
Date : January 8, 2021.2020
Mathematics Subject Classification.
Primary 13J30, 16K40; Secondary 15A22, 26C15, 16W10.
Key words and phrases.
Noncommutative rational function, free skew field, Hilbert’s 17th problem,Positivstellensatz, linear matrix inequality, spectrahedron, linear matrix pencil. ‹ Supported by the NSF grant DMS 1954709.
Let x “ p x , . . . , x d q be freely noncommuting variables. The free algebra C ă x ą of noncommutative polynomials admits a universal skew field of fractions C pă x qą , alsocalled the free skew field [Coh95, CR99], whose elements are noncommutative rationalfunctions. We endow C pă x qą with the unique involution ˚ that fixes the variables andconjugates the scalars. One can consider positivity of noncommutative rational functionson tuples of hermitian matrices. For example, let r “ x ` x ´ p x x ` x x qp x ` x q ´ p x x ` x x q P C pă x qą . It turns out r p X q is a positive semidefinite matrix for every tuple of hermitian matrices X “ p X , X , X , X q belonging to the domain of r (meaning ker X X ker X “ t u inthis particular case). One way to certify this is by observing that r “ r r ˚ ` r r ˚ where r “ p x ´ x x ´ x q x p x ` x q ´ x , r “ p x ´ x x ´ x qp ` x x ´ x q ´ . The solution of Hilbert’s 17th problem in the free skew field presented in this paper(Corollary 5.4) states that every r P C pă x qą , positive semidefinite on its hermitian domain,is a sum of hermitian squares in C pă x qą . This statement was proved in [KPV17] fornoncommutative rational functions r that are regular , meaning that r p X q is well-definedfor every tuple of hermitian matrices. As with most noncommutative Positivstellens¨atze,at the heart of this result is a variation of the Gelfand-Naimark-Segal (GNS) construction.Namely, if r P C pă x qą is not a sum of hermitian squares, one can construct a tuple offinite-dimensional hermitian operators Y that is a sensible candidate for witnessing non-positive-definiteness of r . However, the construction itself does not guarantee that Y actually belongs to the domain of r . This is not a problem if one assumes that r is regular,as it was done in [KPV17]. However, it is worth mentioning that deciding regularity of anoncommutative rational function is a challenge on its own, as observed in [KPV17]. Inthis paper, the domain issue is resolved with an extension result: the tuple Y obtainedfrom the GNS construction can be extended to a tuple of finite-dimensional hermitianoperators in the domain of r without losing the desired features of Y .The first main theorem of this paper pertains to linear matrix pencils and is key forthe extension mentioned above. It might also be of independent interest in the study ofquiver representations and semi-invariants [Kin94, DM17]. Let b denote the Kroneckerproduct of matrices. Theorem A.
Let Λ P M e p C q d be such that Λ b X ` ¨ ¨ ¨ ` Λ d b X d is invertible for some X P M k p C q d . If Y P M ℓ p C q d , Y P M m ˆ ℓ p C q d , Y P M ℓ ˆ m p C q d are such that Λ b ˜ Y Y ¸ ` ¨ ¨ ¨ ` Λ d b ˜ Y d Y d ¸ and Λ b ´ Y Y ¯ ` ¨ ¨ ¨ ` Λ d b ´ Y d Y d ¯ have full rank, then there exists Z P M n p C q d for some n ě m such that Λ b ¨˚˝ Y Y Y Z ˛‹‚ ` ¨ ¨ ¨ ` Λ d b ¨˚˝ Y d Y d Y d Z d ˛‹‚ is invertible. ILBERT’S 17TH PROBLEM IN FREE SKEW FIELDS 3
See Theorem 3.3 for the proof. Together with a truncated rational imitation of theGNS construction, Theorem A leads to a rational Positivstellensatz on free spectrahedra.Given a monic hermitian pencil L “ I ` H x ` ¨ ¨ ¨ ` H d x d , the associated free spec-trahedron D p L q is the set of hermitian tuples X satisfying the linear matrix inequality L p X q ľ
0. Since every convex solution set of a noncommutative polynomial is a freespectrahedron [HM12], the following statement is called a rational convex Positivstel-lensatz, and it generalizes its analogs in the polynomial context [HKM12] and regularrational context [Pas18].
Theorem B.
Let L be a hermitian monic pencil and r P C pă x qą . Then r ľ on D p L q X dom r if and only if r belongs to the rational quadratic module generated by L : r “ r ˚ r ` ¨ ¨ ¨ ` r ˚ m r m ` v ˚ L v ` ¨ ¨ ¨ ` v ˚ n L v n where r i P C pă x qą and v j are vectors over C pă x qą . A more precise quantitative version is given in Theorem 5.2 and has several conse-quences. The solution of Hilbert’s 17th problem in C pă x qą is obtained by taking L “ r P C pă x qą can be represented by a formalrational expression that is well-defined at every hermitian tuple in the domain of r (Propo-sition 2.1); this statement fails in general if arbitrary matrix tuples are considered. Onthe other hand, a Nullstellensatz for cancellation of non-hermitian singularities is givenin Proposition 6.3. Acknowledgment.
The author thanks Igor Klep for valuable comments and suggestionswhich improved the presentation of this paper.2.
Preliminaries
In this section we establish terminology, notation and preliminary results on noncom-mutative rational functions that are used throughout the paper. Let M m ˆ n p C q denotethe space of complex m ˆ n matrices, and M n p C q “ M n ˆ n p C q . Let H n p C q denote the realspace of hermitian n ˆ n matrices. For X “ p X , . . . , X d q P M m ˆ n p C q d , A P M p ˆ m p C q and B P M n ˆ q p C q we write AXB “ p AX B, . . . , AX d B q P M p ˆ q p C q d , X ˚ “ p X ˚ , . . . , X ˚ d q P M n ˆ m p C q d . J. VOL ˇCI ˇC
Free skew field.
We define noncommutative rational functions using formal ra-tional expressions and their matrix evaluations as in [K-VV12].
Formal rational ex-pressions are syntactically valid combinations of scalars, freely noncommuting variables x “ p x , . . . , x d q , rational operations and parentheses. More precisely, a formal rationalexpression is an ordered (from left to right) rooted tree whose leaves have labels from C Y t x , . . . , x d u , and every other node is either labeled ` or ˆ and has two children,or is labeled ´ and has one child. For example, pp ` x q ´ x q x ´ is a formal rationalexpression corresponding to the ordered tree2 x ` ´ x ˆ ´ x ˆ A subexpression of a formal rational expression r is any formal rational expressionwhich appears in the construction of r (i.e., as a sub-tree). For example, all subexpressionsof pp ` x q ´ x q x ´ are2 , x , ` x , p ` x q ´ , x , p ` x q ´ x , x ´ , pp ` x q ´ x q x ´ . Given a formal rational expression r and X P M n p C q d , the evaluation r p X q is definedin the natural way if all inverses appearing in r exist at X . The set of all X P M n p C q d such that r is defined at X is denoted dom n r . The (matricial) domain of r isdom r “ ď n P N dom n r. Note that dom n r is a Zariski open set in M n p C q d for every n P N . A formal rationalexpression r is non-degenerate if dom r ‰ H ; let R C p x q denote the set of all non-degenerate formal rational expressions. On R C p x q we define an equivalence relation r „ r if and only if r p X q “ r p X q for all X P dom r X dom r . Equivalence classes withrespect to this relation are called noncommutative rational functions . By [K-VV12,Proposition 2.2] they form a skew field denoted C pă x qą , which is the universal skew fieldof fractions of the free algebra C ă x ą by [Coh95, Section 4.5]. The equivalence class of r P R C p x q is denoted r P C pă x qą ; we also write r P r and say that r is a representative ofthe noncommutative rational function r .There is a unique involution ˚ on C pă x qą that is determined by α ˚ “ α for α P C and x ˚ j “ x j for j “ , . . . , d . Furthermore, this involution lifts to an involutive map ˚ on theset R C p x q : in terms of ordered trees, ˚ transposes a tree from left to right and conjugatesthe scalar labels. Note that X P dom r implies X ˚ P dom r ˚ for r P R C p x q .2.2. Hermitian domain.
For r P R C p x q let hdom n r “ dom n r X H n p C q d . Thenhdom r “ ď n P N hdom n r ILBERT’S 17TH PROBLEM IN FREE SKEW FIELDS 5 is the hermitian domain of r . Note that hdom n r is Zariski dense in dom n r becauseH n p C q is Zariski dense in M n p C q and dom n r is Zariski open in M n p C q d . Finally, we definethe (hermitian) domain of a noncommutative rational function: for r P C pă x qą letdom r “ ď r P r dom r, hdom r “ ď r P r hdom r. By the definition of the equivalence relation on non-degenerate expressions, r has awell-defined evaluation at X P dom r , written as r p X q , which equals r p X q for any repre-sentative r of r that has X in its domain. The following proposition is a generalizationof [KPV17, Proposition 3.3] and is proved in Subsection 6.1. Proposition 2.1.
For every r P C pă x qą there exists r P r such that hdom r “ hdom r .Remark . There are noncommutative rational functions such that dom r ‰ dom r forevery r P r , see Example 6.2 or [Vol17, Example 3.13].2.3. Linear representation of a formal rational expression.
A fundamental tool forhandling noncommutative rational functions are linear representations (also linearizationsor realizations) [CR99, Coh95, HMS18]. Let r P R C p x q . By [HMS18, Theorem 4.2and Algorithm 4.3] there exist e P N , vectors u, v P C e and an affine matrix pencil M “ M ` M x ` ¨ ¨ ¨ ` M d x d with M j P M e p C q satisfying the following. For every unital C -algebra A and a P A d ,(i) if r can be evaluated at a , then M p a q P GL e p A q and r p a q “ u ˚ M p a q ´ v ;(ii) if M p a q P GL e p A q and A “ M n p C q for some n P N , then r can be evaluated at a .We say that the triple p u, M, v q is a linear representation of r of size e . Usually,linear representations are defined for noncommutative rational functions and with lessemphasis on domains; however, the definition above is more convenient for the purposeof this paper. Remark . In the definition of a linear representation, (ii) is valid not only for M n p C q but more broadly for stably finite algebras [HMS18, Lemma 5.2]. However, it may fail ingeneral, e.g. for the algebra of all bounded operators on an infinite-dimensional Hilbertspace.We will also require the following proposition on pencils that is a combination ofvarious existing results. Proposition 2.4 ([Coh95, K-VV12, DM17]) . Let M be an affine pencil of size e . Thefollowing are equivalent:(i) M P GL e p C pă x qąq ;(ii) there are n P N and X P M n p C q d such that det M p X q ‰ ;(iii) for every n ě e ´ there is X P M n p C q d such that det M p X q ‰ ;(iv) if U P M e ˆ e p C q , V P M e ˆ e p C q satisfy U M V “ , then rk U ` rk V ď e . J. VOL ˇCI ˇC
Proof. (i) ô (ii) follows by the construction of the free skew field via matrix evaluations(cf. [K-VV12, Proposition 2.1]). (iii) ñ (ii) is trivial and (ii) ñ (iii) holds by [DM17,Theorem 1.8]. (iv) ô (i) follows from [Coh95, Corollaries 4.5.9 and 6.3.6] since the freealgebra C ă x ą is a free ideal ring [Coh95, Theorem 5.4.1]. (cid:3) An affine matrix pencil is full [Coh95, Section 1.4] if it satisfies the (equivalent)properties in Proposition 2.4.
Remark . If r P R C p x q admits a linear representation of size e , then hdom n r ‰ H for n ě e ´ n r in dom n r .3. An extension theorem
An affine matrix pencil M of size e is irreducible if U M V “ U P M e ˆ e p C q and V P M e ˆ e p C q implies rk U ` rk V ď e ´
1. In other words, a pencilis not irreducible if it can be put into a 2 ˆ p ‹ ‹ ‹ q by a left and a right basis change. Every irreducible pencil is full.On the other hand, every full pencil is, up to a left and a right basis change, equal toa block upper-triangular pencil whose diagonal blocks are irreducible pencils. In termsof quiver representations [Kin94], M “ M ` ř dj “ M j x j is full/irreducible if and only ifthe p e, e q -dimensional representation p M , M , . . . , M d q of the p d ` q -Kronecker quiver is p , ´ q -semistable/stable.For the purpose of this section we extend evaluations of linear matrix pencils totuples of rectangular matrices. If Λ “ ř dj “ Λ j x j is of size e and X P M ℓ ˆ m p C q d thenΛ p X q “ d ÿ j “ Λ j b X j P M eℓ ˆ em p C q . The following lemma and proposition rely on an ampliation trick in a free algebra todemonstrate the existence of specific invertible evaluations of full pencils (see [HKV20,Section 2.1] for another argument involving such ampliations).
Lemma 3.1.
Let Λ “ ř dj “ Λ j x j be a homogeneous irreducible pencil of size e . Let ℓ ď m and denote n “ p m ´ ℓ qp e ´ q . Given C P M me ˆ ℓe p C q , consider the pencil r Λ ofsize p m ` n q e in d p m ` n qp n ` m ´ ℓ q variables z jpq , r Λ “ ˜ C
00 0 ¸ ` d ÿ j “ n ` m ´ ℓ ÿ q “ ˜ m ÿ p “ ˜ p E p,q b Λ j ¸ z jpq ` m ` n ÿ p “ m ` ˜ q E p ´ m,q b Λ j ¸ z jpq ¸ where p E p,q P M m ˆp n ` m ´ ℓ q p C q and q E p ´ m,q P M n ˆp n ` m ´ ℓ q p C q are the standard matrix units.If C has full rank, then the pencil r Λ is full.Proof. Suppose U and V are constant matrices with e p m ` n q columns and e p m ` n q rows,respectively, that satisfy U r Λ V “
0. There is nothing to prove if U “
0, so let U ‰ ILBERT’S 17TH PROBLEM IN FREE SKEW FIELDS 7
Write U “ ´ U ¨ ¨ ¨ U m ` n ¯ , V “ ¨˚˚˚˚˝ V V ... V n ` m ´ ℓ ˛‹‹‹‹‚ where each U p has e columns, V has ℓe rows, and each V q with q ą e rows. Alsolet U “ p U ¨ ¨ ¨ U m q . Then U r Λ V “ U CV “ , (3.1) U p Λ V q “ ď p ď m ` n, ď q ď n ` m ´ ℓ. (3.2)Since C has full rank, (3.1) implies rk U ` rk V ď me . Note that U p ‰ ď p ď m ` n because U ‰
0. Since Λ is irreducible and U p ‰ p , (3.2)implies rk V q ď e ´ U p ` rk V q ď e ´ p, q ą
0. Thenrk U ` rk V ď rk U ` rk V ` m ` n ÿ p “ m ` rk U p ` n ` m ´ ℓ ÿ q “ rk V q ď me ` n p e ´ q ` p m ´ ℓ qp e ´ q“ p m ` n q e by the choice of n . Therefore r Λ is full. (cid:3)
Proposition 3.2.
Let Λ be a homogeneous full pencil of size e , and let X P M m ˆ ℓ p C q d with ℓ ď m be such that Λ p X q has full rank. Then there exist p X P M m ˆp n ` m ´ ℓ q p C q d and q X P M n ˆp n ` m ´ ℓ q p C q d for some n P N such that (3.3) det Λ ˜ X p X q X ¸ ‰ . Proof.
A full pencil is up to a left-right basis change equal to a block upper-triangularpencil with irreducible diagonal blocks. Suppose that the lemma holds for irreduciblepencils; since the set of pairs p p X, q X q P M m ˆp n ` m ´ ℓ q p C q d ˆ M n ˆp n ` m ´ ℓ q p C q d satisfying(3.3) is Zariski open, the lemma then also holds for full pencils. Thus we can withoutloss of generality assume that Λ is irreducible.Let n “ p m ´ ℓ qp e ´ q and e “ p m ` n q e . By Lemma 3.1 applied to C “ ř dj “ X j b Λ j and Proposition 2.4, there exists Z P M e ´ p C q d p m ` n qp n ` m ´ ℓ q such that r Λ p Z q is invertible. Therefore the matrix d ÿ j “ Λ j b ˜` I b X j
00 0 ˘ ` n ` m ´ ℓ ÿ q “ ˜ n ÿ p “ ´ Z jpq b p E p,q ¯ ` m ` n ÿ p “ m ` ´ Z jpq b q E p ´ m,q ¯¸¸ is invertible since it is similar to r Λ p Z q (via a permutation matrix). Thus there are p X P M m ˆp n ` m ´ ℓ q p C q d and q X P M n ˆp n ` m ´ ℓ q p C q d such thatdet Λ ˜ X p X q X ¸ ‰ J. VOL ˇCI ˇC where n “ p e ´ q m ` n p e ´ q . (cid:3) We are ready to prove the first main result of the paper.
Theorem 3.3.
Let Λ be a full pencil of size e , and let Y P M ℓ p C q d , Y P M m ˆ ℓ p C q d , Y P M ℓ ˆ m p C q d be such that (3.4) Λ ˜ YY ¸ , Λ ´ Y Y ¯ have full rank. Then there are n ě m and Z P M n p C q d such that det Λ ¨˚˝ Y Y Y Z ˛‹‚ ‰ . Proof.
By Proposition 3.2 and its transpose analog there exist k P N and A P M ℓ ˆp k ` m ´ ℓ q p C q d , B P M m ˆp k ` m ´ ℓ q p C q d , C P M k ˆp k ` m ´ ℓ q p C q d ,A P M p k ` m ´ ℓ qˆ ℓ p C q d , B P M p k ` m ´ ℓ qˆ m p C q d , C P M p k ` m ´ ℓ qˆ k p C q d such that the matrices Λ ¨˚˝ Y A Y B C ˛‹‚ , Λ ˜ Y Y A B C ¸ are invertible. Consequently there exists ε P C zt u such that ¨˚˚˚˚˚˝ Λ p Y q ˛‹‹‹‹‹‚ ` ε ¨˚˚˚˚˚˚˝ ˜ ¸ Λ ˜ Y Y A B C ¸ Λ ¨˚˝ Y A Y B C ˛‹‚ ¨˚˝ ˛‹‚ ˛‹‹‹‹‹‹‚ is invertible; this matrix is similar to(3.5) Λ ¨˚˚˚˚˚˝ Y εY εY
00 0 εA εB εC εY εA εY εB εC ˛‹‹‹‹‹‚ . Thus the matrix (3.5) is invertible; its block structure and the linearity of Λ imply that(3.5) is invertible for every ε ‰
0, so we can choose ε “
1. After performing elementary
ILBERT’S 17TH PROBLEM IN FREE SKEW FIELDS 9 row and column operations on (3.5) we conclude that(3.6) Λ ¨˚˚˚˚˚˝
Y Y Y ´ Y B ´ Y ´ Y A B A C C ˛‹‹‹‹‹‚ is invertible. So the lemma holds for n “ p m ` k q . (cid:3) Remark . It follows from the proofs of Proposition 3.2 and Theorem 3.3 that one canchoose n “ ` e m ` em p eℓ ´ q ` ℓ p eℓ ´ q ˘ in Theorem 3.3. However, this is unlikely to be the minimal choice for n .Let M p C q be the algebra of N ˆ N matrices over C that have only finitely manynonzero entries in each column; that is, elements of M p C q can be viewed as operatorson ‘ N C . Given r P R C p x q let dom r be the set of tuples X P M p C q d such that r p X q iswell-defined. If p u, M, v q is a linear representation of r of size e , then M p X q P M e p M p C qq is invertible for every X P dom r by the definition of a linear representation adopted inthis paper. Proposition 3.5.
Let r P R C p x q . If X P H ℓ p C q d and Y P M m ˆ ℓ p C q d are such that ¨˚˝ X Y ˚ Y W ˛‹‚ P dom r for some W P M p C q d , then there exist n ě m , E P M n p C q and Z P H n p C q d such that ¨˚˚˝ X ´ Y ˚ ¯ E ˚ E ˜ Y ¸ Z ˛‹‹‚ P hdom r. Proof.
Let p u, M, v q be a linear representation of r of size e . By assumption, M ¨˚˝ X Y ˚ Y W ˛‹‚ is an invertible matrix over M p C q . If M “ M ` M x ` ¨ ¨ ¨ ` M d x d , then the matrices M b ˜ I ¸ ` d ÿ j “ M b ˜ X j Y j ¸ , M b ´ I ¯ ` d ÿ j “ M b ´ X j Y ˚ j ¯ have full rank. Let n P N be as in Theorem 3.3. Then there is Z P M n p C q ` d such that(3.7) det ¨˚˝ M b ¨˚˝ I Z ˛‹‚ ` d ÿ j “ M b ¨˚˝ X j Y ˚ j Y j Z j ˛‹‚˛‹‚ ‰ is invertible. The set of all Z P M n p C q ` d satisfying (3.7) is thus a nonempty Zariskiopen set in M n p C q ` d . Since the set of positive definite n ˆ n matrices is Zariski dense inM n p C q , there exists Z P H n p C q ` d with Z ą Z “ E ´ E ´˚ ,let Z j “ EZ j E ˚ for 1 ď j ď d . Then M ¨˚˚˝ X ´ Y ˚ ¯ E ˚ E ˜ Y ¸ Z ˛‹‹‚ is invertible, so ¨˚˚˝ X ´ Y ˚ ¯ E ˚ E ˜ Y ¸ Z ˛‹‹‚ P hdom r by the definition of a linear representation. (cid:3) We also record a non-hermitian version of Proposition 3.5.
Proposition 3.6.
Let r P R C p x q . If X P M m ˆ ℓ p C q d with ℓ ď m is such that ˜ X W ¸ P dom r for some W P M p C q d , then there exist n ě m and Z P M n ˆp n ´ ℓ q p C q d such that ˜ X Z ¸ P dom r. Proof.
We apply a similar reasoning as in the proof of Proposition 3.5, only this timewith Proposition 3.2 instead of Theorem 3.3, and without hermitian considerations. (cid:3) Multiplication operators attached to a formal rational expression
In this section we assign a tuple of operators X on a vector space of countabledimension to each formal rational expression r , so that r is well-defined at X and thefinite-dimensional restrictions of X partially retain a certain multiplicative property.Fix an expression r P R C p x q , and let R “ t u Y t q P R C p x qz C : q is a subexpression of r or r ˚ u Ă R C p x q . Note that R is finite, hdom q Ě hdom r for q P R , and q P R implies q ˚ P R . Let R Ă C pă x qą be the set of noncommutative rational functions represented by R . For ℓ P N we define finite-dimensional vector subspaces V ℓ “ span C ℓ hkkikkj R ¨ ¨ ¨ R Ă C pă x qą . Note that V ℓ Ď V ℓ ` since 1 P R . Furthermore, let V “ Ť ℓ P N V ℓ . Then V is a finitelygenerated ˚ -subalgebra of C pă x qą . For j “ , . . . , d we define operators X j : V Ñ V, X j s “ x j s . ILBERT’S 17TH PROBLEM IN FREE SKEW FIELDS 11
Lemma 4.1.
There is a linear functional φ : V Ñ C such that φ p s ˚ q “ φ p s q and φ p ss ˚ q ą for all s P V zt u .Proof. For some X P hdom r let m “ max q P R } q p X q} . Let ℓ P N . Since V ℓ is finite-dimensional, there exist n ℓ P N and X p ℓ q P hdom n ℓ r such that(4.1) max q P R } q p X p ℓ q q} ď m ` s p X p ℓ q q ‰ s P V ℓ zt u by the local-global linear dependence principle for noncommutative rational functions,see [Vol18, Theorem 6.5] or [BPT13, Corollary 8.87]. Define φ : V Ñ C , φ p s q “ ÿ ℓ “ ℓ ! ¨ n ℓ tr ` s p X p ℓ q q ˘ . Since V is a C -algebra generated by R , routine estimates show that φ is well-defined. Itis also clear that φ has the desired properties. (cid:3) For the rest of the paper fix a functional φ as in Lemma 4.1. Then(4.2) p s , s q “ φ p s ˚ s q is an inner product on V . With respect to this inner product we can inductively build anordered orthogonal basis B of V with the property that B X V ℓ is a basis of V ℓ for every ℓ P N . Lemma 4.2.
With respect to the inner product (4.2) and the ordered basis B as above,operators X , . . . , X d are represented by hermitian matrices in M p C q , and X P dom r .Proof. Since ` X j s , s ˘ “ φ ` s ˚ x j s ˘ “ ` s , X j s ˘ for all s , s P V and X j p V ℓ q Ď V ℓ ` for all ℓ P N , it follows that the matrix representationof X j with respect to B is hermitian and has only finitely many nonzero entries in eachcolumn and row. The rest follows inductively on the construction of r since X j are theleft multiplication operators on V . (cid:3) Next we define a complexity-measuring function τ : R C p x q Ñ N Y t u as in [KPV17,Section 4]:(i) τ p α q “ α P C ;(ii) τ p x j q “ ď j ď d ;(iii) τ p s ` s q “ max t τ p s q , τ p s qu for s , s P R C p x q ;(iv) τ p s s q “ τ p s q ` τ p s q for s , s P R C p x q ;(v) τ p s ´ q “ τ p s q for s, s ´ P R C p x q .Note that τ p s ˚ q “ τ p s q for all s P R C p x q . Proposition 4.3.
Let the notation be as above, and let U be a finite-dimensional Hilbertspace containing V ℓ ` . If X is a d -tuple of hermitian operators on U such that X P hdom r and X j | V ℓ “ X j | V ℓ for j “ , . . . , d , then X P hdom q and (4.3) q p X q s “ qs for every q P R and s P ℓ hkkikkj R ¨ ¨ ¨ R satisfying τ p q q ` τ p s q ď ℓ ` .Proof. First note that for every s P R ¨ ¨ ¨ R ,(4.4) τ p s q ď k ñ s P k hkkikkj R ¨ ¨ ¨ R since τ ´ p q “ C and R X C “ t u . We prove (4.3) by induction on the construction of q . If q “ q “ x j then τ p s q ď ℓ so (4.3) holds by (4.4).Next, if (4.3) holds for q , q P R such that q ` q P R or q q P R , then it also holds forthe latter by the definition of τ and (4.4). Finally, suppose that (4.3) holds for q P R zt u and assume q ´ P R . If 2 τ p q ´ q ` τ p s q ď ℓ `
2, then 2 τ p q q ` p τ p q ´ q ` τ p s qq ď ℓ `
2. Inparticular, τ p q ´ s q ď ℓ and so q ´ s P ℓ hkkikkj R ¨ ¨ ¨ R by (4.4). Therefore q p X q q ´ s “ qq ´ s “ s by the induction hypothesis, and hence q ´ p X q s “ q ´ s since X P hdom q ´ . Thus (4.3)holds for q ´ . (cid:3) Positive noncommutative rational functions
In this section we prove various positivity statements for noncommutative rationalfunctions. Let L be a hermitian monic pencil of size e ; that is, L “ I ` H x ` ¨ ¨ ¨ ` H d x d with H j P H e p C q . Then D p L q “ ď n P N D n p L q , where D n p L q “ t X P H n p C q d : L p X q ľ u , is a free spectrahedron . The main result of the paper is Theorem 5.2, which describesnoncommutative rational functions that are positive semidefinite or undefined at eachtuple in a given free spectrahedron D p L q . In particular, Theorem 5.2 generalizes [Pas18,Theorem 3.1] to noncommutative rational functions with singularities in D p L q .5.1. Rational convex Positivstellensatz.
Let L be a hermitian monic pencil of size e .To r P R C p x q we assign the finite set R , vector spaces V ℓ and operators X j as in Section4. For ℓ P N we also define S ℓ “ t s P V ℓ : s “ s ˚ u Q ℓ “ i s ˚ i s i ` ÿ j v ˚ j L v j : s i P V ℓ , v j P V eℓ + Ă S ℓ ` . Then S ℓ is a real vector space and Q ℓ is a convex cone. The proof of the following propo-sition is a rational modification of a common argument in free real algebraic geometry(cf. [HKM12, Proposition 3.1] and [KPV17, Proposition 4.1]). A convex cone is salientif it does not contain a line. ILBERT’S 17TH PROBLEM IN FREE SKEW FIELDS 13
Proposition 5.1.
The cone Q ℓ is salient and closed in S ℓ ` with the Euclidean topology.Proof. As in the proof of Lemma 4.1 there exists X P hdom r such that s p X q ‰ s P V ℓ ` zt u . Furthermore, we can choose X close enough to 0 so that L p X q ľ I . Then clearly s p X q ľ s P Q ℓ , so Q ℓ X ´ Q ℓ “ t u and thus Q ℓ is salient. Note that } s } ‚ “ } s p X q} is a norm on V ℓ ` . Also, finite-dimensionality of S ℓ ` implies that everyelement of Q ℓ can be written as a sum of N “ ` dim S ℓ ` elements of the form s ˚ s and v ˚ L v for s P V ℓ , v P V eℓ by Carath´eodory’s theorem [Bar02, Theorem I.2.3]. Assume that a sequence t r n u n Ă Q ℓ converges to s P S ℓ ` . After restricting to a subsequence we can assume that there is0 ď M ď N such that r n “ M ÿ i “ s ˚ n,i s n,i ` N ÿ j “ M ` v ˚ n,j L v n,j for all n P N . The definition of the norm } ¨ } ‚ implies } s n i } ‚ ď } r n } ‚ and max ď i ď e }p v n j q i } ‚ ď } r n } ‚ . In particular, the sequences t s n,i u n Ă V ℓ for 1 ď i ď M and t v n,j u n Ă V eℓ for 1 ď j ď N are bounded. Hence, after restricting to subsequences, we may assume that they areconvergent: s i “ lim n s n,i for 1 ď i ď M and v j “ lim n v n,j for 1 ď j ď N . Consequentlywe have s “ lim n r n “ M ÿ i “ s ˚ i s i ` N ÿ j “ M ` v ˚ j L v j P Q ℓ . (cid:3) We are now ready to prove the main result of this paper by combining a truncatedGNS construction with extending matrix tuples into the domain of a rational expressionas in Proposition 3.5.
Theorem 5.2 (Rational convex Positivstellensatz) . Let L be a hermitian monic penciland r P R C p x q . If Q τ p r q` is as above, then r p X q ľ for every X P hdom r X D p L q ifand only if r P Q τ p r q` .Proof. Only the forward implication is nontrivial. Let ℓ “ τ p r q ´
2. If r ‰ r ˚ , thenthere exists X P hdom r such that r p X q ‰ r p X q ˚ . Thus we assume r “ r ˚ . Suppose that r R Q ℓ ` . Since Q ℓ ` is a salient closed convex cone in S ℓ ` by Proposition 5.1, thereexists a linear functional λ : S ℓ ` Ñ R such that λ p Q ℓ ` zt uq “ R ą and λ p r q ă λ to λ : V ℓ ` Ñ C as λ p s q “ λ p s ` s ˚ q ` i λ p i p s ˚ ´ s qq . Then x s , s y “ λ p s ˚ s q defines ascalar product on V ℓ ` . Recall that X j p V ℓ ` q Ď V ℓ ` . Then for s P V ℓ ` and s P V ℓ ` ,(5.1) x X j s , s y “ λ ` s ˚ x j s ˘ “ x s , X j s y . Furthermore,(5.2) x L p X q v , v y “ λ p v ˚ L v q ą for all v P V eℓ ` , where the canonical extension of x¨ , ¨y to a scalar product on C e b V ℓ ` is considered.Let B be an ordered orthogonal basis of V with respect to the inner product p¨ , ¨q as in Section 4; recall that such a basis has the property that B X V k is a basis for V k for all k P N . Let B be an ordered orthogonal basis of V ℓ ` with respect to x¨ , ¨y thatcontains a basis for V ℓ ` , and let B “ B z V ℓ ` . If we identify operators X j with theirmatrix representations relative to the ordered basis p B , B q of V , then X j P M p C q arehermitian matrices by Lemma 4.2 and (5.1).Let U be the orthogonal complement of V ℓ ` in V ℓ ` relative to x¨ , ¨y . Since X j p V ℓ ` q Ď V ℓ ` , we can consider the restriction X j | V ℓ ` in a block form ˜ X j Y j ¸ with respect to the decomposition V ℓ ` “ V ℓ ` ‘ U . Since X P dom r , by Proposition3.5 there exist a finite-dimensional vector space U , a scalar product on V ℓ ` ‘ U ‘ U extending x¨ , ¨y , an operator E on U ‘ U , and a d -tuple Z of hermitian operators on U ‘ U such that(5.3) r X : “ ¨˚˚˝ X ´ Y ˚ ¯ E ˚ E ˜ Y ¸ Z ˛‹‹‚ P hdom r. Since X j p V ℓ q Ď V ℓ ` , we conclude that(5.4) r X j | V ℓ “ X j | V ℓ . Observe that for all but finitely many ε , ε ą Z, E with ε Z, ε E and(5.3) still holds. By (5.2) we can thus assume that Z and E are close enough to 0 so that L p r X q ľ
0. Finally, since (5.4) holds and 2 τ p r q ` τ p q “ ℓ `
2, Proposition 4.3 implies x r p r X q , y “ x r , y “ λ p r q ă . Therefore r X P hdom r X D p L q and r p r X q is not positive semidefinite. (cid:3) Given a unital ˚ -algebra A and A “ A ˚ P M ℓ p A q , the quadratic module in A generated by A is QM A p A q “ j v ˚ j p ‘ A q v j : v j P A ℓ ` + . Theorem 5.2 then in particular states that noncommutative rational functions positivesemidefinite on a free spectrahedron D p L q belong to QM C pă x qą p L q . Remark . Let r P R C p x q and n “ ` e m ` em p eℓ ´ q ` ℓ p eℓ ´ q ˘ where ℓ “ dim V τ p r q´ , m “ dim V τ p r q ´ dim V τ p r q´ and e is the size of a linear represen-tation of r . If r ń r X D p L q , then by Remark 3.4 and the proofs of Theorem5.2 and Proposition 3.5 there exists X P hdom n r X D n p L q such that r p X q ń ILBERT’S 17TH PROBLEM IN FREE SKEW FIELDS 15
The solution of Hilbert’s 17th problem for a free skew field is now as follows.
Corollary 5.4.
Let r P C pă x qą . Then r ľ on hdom r if and only if r “ r r ˚ ` ¨ ¨ ¨ ` r m r ˚ m for some r i P C pă x qą with hdom r i Ě hdom r .Proof. By Proposition 2.1 there exists r P r such that hdom r “ hdom r . The corollarythen follows directly from Theorem 5.2 applied to L “ V τ p r q contains hdom r . (cid:3) Remark . Corollary 5.4 also indicates a subtle distinction between solutions of Hilbert’s17th problem in the classical commutative context and in the free context. While ev-ery (commutative) positive rational function ρ is a sum of squares of rational functions,in general one cannot choose summands that are defined on the whole real domain ofthe original function ρ . On the other hand, a positive noncommutative rational func-tion always admits a sum-of-squares representation with terms defined on its hermitiandomain.For a possible future use we describe noncommutative rational functions whose in-vertible evaluations have nonconstant signature; polynomials of this type were of interestin [HKV20, Section 3.3]. Corollary 5.6.
Let r “ r ˚ P C pă x qą . The following are equivalent:(i) there are n P N and X, Y P hdom n r such that r p X q , r p Y q are invertible and havedistinct signatures;(ii) neither r or ´ r equals ř i r i r ˚ i for some r i P C pă x qą .Proof. (i) ñ (ii) If ˘ r “ ř i r i r ˚ i , then ˘ r p X q ľ X P hdom r .(ii) ñ (i) Let O n “ hdom n r X hdom n r ´ . By Remark 2.5 there is n P N such that O n ‰ H for all n ě n . Suppose that r has constant signature on O n for each n ě n ,i.e., r p X q has π n positive eigenvalues for every X P O n . Since O k ‘ O ℓ Ă O k ` ℓ for all k, ℓ P N , we have(5.5) nπ m “ π mn “ mπ n for all m, n ě n . If π n “ n for some n ě n , then π n “ n for all n ě n by (5.5), so r ľ O n for every n . Thus r “ ř i r i r ˚ i by Theorem 5.2. Analogous conclusion holdsif π n “ n ě n . However, (5.5) excludes any alternative: if n ď m ă n and n is a prime number, then 0 ă π n ă n contradicts (5.5). (cid:3) Positivity and invariants.
Let G be a subgroup of the unitary group U d p C q . Theaction of G on C d induces a linear action of G on C pă x qą . If G is finite and solvable,then the subfield of G -invariants C pă x qą G is finitely generated [KPPV20, Theorem 1.1]and in many cases again a free skew field [KPPV20, Theorem 1.3]. Furthermore, wecan now extend [KPPV20, Corollary 6.6] to invariant noncommutative rational functionswith singularities. Corollary 5.7.
Let G Ă U d p C q be a finite solvable group. Then there exists R G P GL | G | p C pă x qąq with the following property. If r P C pă x qą G and L is a hermitian monicpencil of size e , then r ľ on hdom r X D p L q if and only if r P QM C pă x qą G p L G q , where L G “ R ˚ G R G ‘ p R G b I q ˚ ˜à g P G L g ¸ p R G b I q P M | G |p e ` q p C pă x qą G q . Proof.
Combine [KPPV20, Corollary 6.4] and Theorem 5.2. (cid:3)
Real free skew field and other variations.
In this subsection we explain howthe preceding results apply to real free skew fields and their symmetric evaluations, andto another natural involution on a free skew field.
Corollary 5.8 (Real version of Theorem 5.2) . Let L be a symmetric monic pencil ofsize e and r P R pă x qą . Then r p X q ľ for every X P hdom r X D p L q if and only if r P QM R pă x qą p L q .Proof. If r P R pă x qą and r ľ r X D p L q , then r P QM C b R pă x qą p L q by Theorem 5.2because the complex vector spaces V ℓ are spanned with functions given by subexpressionsof some r P r , and we can choose r in which only real scalars appear. For s P C b R pă x qą we define re p s q “ p s ` s q and im p s q “ i p s ´ s q in R pă x qą . If r “ ÿ j s ˚ j s j ` ÿ k v ˚ k L v k for s j P p C b R pă x qąq and v k P p C b R pă x qąq e , then r “ re p r q “ ÿ j p re p s j q ˚ re p s j q ` im p s j q ˚ im p s j qq ` ÿ k p re p v k q ˚ L re p v k q ` im p v k q ˚ L im p v k qq and so r P QM R pă x qą p L q . (cid:3) Given r P R pă x qą one might prefer to consider only the tuples of real symmetricmatrices in the domain of r , and not the whole hdom r . Since there exist ˚ -embeddingsM n p C q ã Ñ M n p R q , evaluations on tuples of real symmetric 2 n ˆ n matrices carry at leastas much information as evaluations on tuples of hermitian n ˆ n matrices. Consequently,all dimension-independent statements in this paper also hold if only symmetric tuplesare considered. However, it is worth mentioning that for r P R pă x qą , it can happen thatdom n r contains no tuples of symmetric matrices for all odd n , e.g. if r “ p x x ´ x x q ´ .Another commonly considered free skew field with involution is C pă x, x ˚ qą , generatedwith 2 d variables x , . . . , x d , x ˚ , . . . , x ˚ d , which is endowed with the involution ˚ that swaps x j and x ˚ j . Elements of C pă x, x ˚ qą can be evaluated on d -tuples of complex matrices.The results of this paper also directly apply to C pă x, x ˚ qą and such evaluations because C pă x, x ˚ qą is freely generated by elements p x j ` x ˚ j q , i p x ˚ j ´ x j q which are fixed by ˚ .Finally, as in Corollary 5.8 we see that a suitable analog of Theorem 5.2 also holds for R pă x, x ˚ qą and evaluations on d -tuples of real matrices. ILBERT’S 17TH PROBLEM IN FREE SKEW FIELDS 17
Examples of non-convex Positivstellens¨atze.
Given m “ m ˚ P M ℓ p C pă x qąq let D p m q “ ď n P N D n p m q , where D n p m q “ t X P hdom n m : m p X q ľ u , be its positivity domain . Here, the domain of m is the intersection of domains of itsentries. Proposition 5.9.
Let m “ m ˚ P M ℓ p C pă x qąq and assume there exist a hermitian monicpencil L of size e ě ℓ , a ˚ -automorphism ϕ of C pă x qą , and A P GL e p C pă x qąq such that (5.6) ϕ p m q ‘ I “ A ˚ LA. If r P C pă x qą , then r ľ on hdom r X D p m q if and only if r P QM C pă x qą p m q .Proof. The relation (5.6), Remark 2.5 and the convexity of D n p L q imply that the sets D n p ϕ p m qq and D n p m q have the same closures as their interiors in the Euclidean topologyfor all but finitely many n . Therefore r | hdom r X D p m q ľ ô ϕ p r q| hdom ϕ p r qX D p ϕ p m qq ľ ô ϕ p r q| hdom ϕ p r qX D p L q ľ ô ϕ p r q P QM C pă x qą p L qô r P QM C pă x qą p ϕ ´ p L qq “ QM C pă x qą p m q by Theorem 5.2 and (5.6). (cid:3) The following example presents a family of quadratic noncommutative polynomials q “ q ˚ P C ă x, x ˚ ą that admit a rational Positivstellensatz on their (not necessarilyconvex) positivity domains D p q q “ t X : q p X, X ˚ q ľ u . Example 5.10.
Given a linearly independent set t a , . . . , a n u Ă span C t , x , . . . , x d u let q “ a ˚ a ´ a ˚ a ´ ¨ ¨ ¨ ´ a ˚ n a n P C ă x, x ˚ ą . One might say that q is a hereditary quadratic polynomial of positive signature 1. Notethat D p q q is not convex if a R C . Since a , . . . , a n are linearly independent affine polyno-mials in C ă x ą (and in particular n ď d ), there exists a linear fractional automorphism ϕ on C pă x qą such that ϕ ´ p x j q “ a j a ´ for 1 ď j ď n . We extend ϕ uniquely to a ˚ -automorphism on C pă x, x ˚ qą . Then ϕ p a q ´˚ ϕ p q q ϕ p a q ´ “ ´ x ˚ x ´ ¨ ¨ ¨ ´ x ˚ n x n and thus ϕ p q q ‘ I n “ A ˚ LA where L “ ¨˚˚˚˚˝ x ˚ ¨ ¨ ¨ x ˚ n x . . .... . . . x n ˛‹‹‹‹‚ , A “ ¨˚˚˚˚˝ ϕ p a q´ x ´ x n ˛‹‹‹‹‚ Therefore r ľ r X D p q q if and only if r P QM C pă x,x ˚ qą p q q for every r P C pă x, x ˚ qą by Proposition 5.9. For example, the polynomial x ˚ x ´ x x ˚ ´ “ p x ´ x ´˚ qp x ˚ ´ x ´ q ` x ´˚ p x ˚ x ´ q x ´ P QM C pă x,x ˚ qą p x ˚ x ´ q . On the other hand, we claim that x x ˚ ´ R QM C ă x,x ˚ ą p x ˚ x ´ q (cf. [HM04, Example4]). If x x ˚ ´ C ă x,x ˚ ą p x ˚ x ´ q , then the implication S ˚ S ´ I ľ ñ SS ˚ ´ I ľ S on an infinite-dimensional Hilbert space; however itfails if S is the forward shift operator on ℓ p N q . A different Positivstellensatz (polynomial,but with a slack variable) for hereditary quadratic polynomials is given in [HKV20,Corollary 4.6].5.5. Eigenvalue optimization.
Theorem 5.2 is also essential for optimization of non-commutative rational functions. Namely, it implies that finding the eigenvalue supremumor infimum of a noncommutative rational function on a free spectrahedron is equivalentto solving a semidefinite program [BPT13]. This equivalence was stated in [KPV17, Sec-tion 5.2.1] for regular noncommutative rational functions; the novelty is that Theorem5.2 now confirms its validity for noncommutative rational functions with singularities.Let L be a hermitian monic pencil of size e , and let r “ r ˚ P C pă x qą . Suppose we areinterested in µ ˚ “ sup X P hdom r X D p L q “ maximal eigenvalue of r p X q ‰ . Choose some r P r (the simpler representative the better) and let ℓ “ τ p r q `
1. Theorem5.2 then implies that(5.7) µ ˚ “ inf µ P R : µ ´ f “ M ÿ i “ s ˚ i s i ` N ÿ j “ v ˚ j L v j : s i P V ℓ , v j P V eℓ + where we can take M “ dim S ℓ ` N “ dim S ℓ ` ` L ) eigenvalue supremumof r , one solves the semidefinite programmin H µ subject to µ ´ f “ ~w ˚ H ~w,H ľ H is a p dim V ℓ q ˆ p dim V ℓ q hermitian matrix and ~w is a vectorized basis of V ℓ . Forconstrained eigenvalue optimization ( L is present), one can set up a similar semidefiniteprogram using localizing matrices [BKP16, Definition 1.41]. ILBERT’S 17TH PROBLEM IN FREE SKEW FIELDS 19 More on domains
In this section we prove two new results on (hermitian) domains. One of them isthe aforementioned Proposition 2.1, which states that every noncommutative rationalfunction admits a representative with the largest hermitian domain. The other one isProposition 6.3 on cancellation of singularities of noncommutative rational functions.6.1.
Representatives with the largest hermitian domain.
We will require a techni-cal lemma about matrices over formal rational expressions and their hermitian domains.A representative of a matrix m over C pă x qą is a matrix over R C p x q of representatives of m ij , and the domain of a matrix over R C p x q is the intersection of domains of its entries. Lemma 6.1.
Let m be an e ˆ e matrix over R C p x q such that m P GL e p C pă x qąq . Thenthere exists s P m ´ such that hdom m X hdom m ´ “ hdom s .Proof. We prove the statement by induction on e . If e “
1, then m ´ is the desiredexpression. Assume the statement holds for matrices of size e ´
1, and let c be thefirst column of m . Then hdom c Ě hdom m and c p X q is of full rank for every X P hdom m X hdom m ´ . Hence(6.1) hdom p c ˚ c q ´ Ě hdom m X hdom m ´ . Let p m be the Schur complement of ˚ in m ˚ m . Note that the entries of p m are polyno-mials in entries of m ˚ m and p ˚ q ´ , so there is p m P p m such that(6.2) hdom p m “ hdom m X hdom p c ˚ c q ´ . If X P hdom m , then m p X q is invertible if and only if p c ˚ c qp X q and p m p X q are invertible.Thus by (6.1) and (6.2) we have(6.3) hdom m X hdom m ´ “ hdom m X p hdom p c ˚ c q ´ X hdom p m ´ q . Since ˆ m is an p e ´ q ˆ p e ´ q matrix, by the induction hypothesis there exists p s P p m ´ such that hdom p m X hdom p m ´ “ hdom p s . By (6.3) we have(6.4) hdom m X hdom m ´ “ hdom m X p hdom p c ˚ c q ´ X hdom p s q . The entries of p m ˚ m q ´ can be represented by expressions s ij which are sums and prod-ucts of expressions m ij , m ˚ ij , p c ˚ c q ´ , p s ij . Thus s P p m ˚ m q ´ satisfieshdom m X hdom m ´ “ hdom s by (6.4). Finally, s “ s m ˚ is the desired expression because m ´ “ p m ˚ m q ´ m ˚ . (cid:3) Proof of Proposition 2.1.
Let r P C pă x qą . Let e P N , an affine matrix pencil M of size e and u, v P C e be such that r “ u ˚ M ´ v in C pă x qą , and e is minimal. Recall that dom r “ Ť r P r dom r . By comparing p u, M, v q with linear representations of representatives of r asin [CR99, Theorem 1.4] it follows that(6.5) dom r Ď ď n P N X P M n p C q d : det M p X q ‰ ( . Since M contains no inverses, it is defined at every matrix tuple; thus by Lemma 6.1 thereis a representative of M ´ whose hermitian domain equals t X “ X ˚ : det M p X q ‰ u .Since r is a linear combination of the entries in M ´ , there exists r P r such thathdom r “ hdom r by (6.5). (cid:3) Example 6.2.
The domain of r P C pă x qą given by the expression p x ´ x x ´ x q ´ equalsdom r “ ď n P N X P M n p C q : det ˜ X X X X ¸ ‰ + and dom r Ĺ dom r for every r P r by [Vol17, Example 3.13].Following the proof of Proposition 2.1 and Lemma 6.1 let m “ p x x x x q . Then m ˚ m “ ˜ x ` x x x ` x x x x ` x x x ` x ¸ and the Schur complement of m ˚ m with respect to the p , q -entry equals p m “ x ` x ´ p x x ` x x qp x ` x q ´ p x x ` x x q . Since m ´ “ p m ˚ m q ´ m ˚ “ ˜ ‹ ‹´ p m ´ p x x ` x x qp x ` x q ´ p m ´ ¸ ˜ ‹ x ‹ x ¸ and r “ ´ ¯ m ´ ˜ ¸ we conclude that the formal rational expression ` x ` x ´ p x x ` x x qp x ` x q ´ p x x ` x x q ˘ ´ ` x ´ p x x ` x x qp x ` x q ´ x ˘ represents r , and its hermitian domain coincides with hdom r . Of course, the expression p x ´ x x ´ x q ´ is a much simpler representative of r .6.2. Cancellation of singularities.
In the absence of left ideals in skew fields, thefollowing proposition serves as a rational analog of Bergman’s Nullstellensatz for non-commutative polynomials [HM04, Theorem 6.3]. The proof below omits some of thedetails since it is a derivate of the proof of Theorem 5.2.
Proposition 6.3.
The following are equivalent for r , s P C pă x qą .(i) ker r p X q Ď ker s p X q for all X P dom r X dom s ;(ii) dom p sr ´ q Ě dom r X dom s .Proof. (ii) ñ (i) If (ii) holds, then s p X q “ p s p X q r p X q ´ q r p X q for every X P dom r X dom s ,and so ker r p X q Ď ker s p X q .(i) ñ (ii) Suppose (ii) does not hold; thus there are r P r , s P s and Y P dom r X dom s such that det r p Y q “
0. Similarly as in Section 4 denote R “ t u Y t q P R C p x qz C : q is a subexpression of r or s u ILBERT’S 17TH PROBLEM IN FREE SKEW FIELDS 21 and let R be its image in C pă x qą . We also define finite-dimensional vector spaces V ℓ andthe finitely generated algebra V as before. The left ideal V r in V is proper: if qr “ q P V , then q p Y q r p Y q “ I since Y P dom q , which contradicts det r p Y q “
0. Furthermore, s R V r since (ii) does not hold. Let K “ V { V r , and let K ℓ be the image of V ℓ for every ℓ P N . Let X j : K Ñ K be the operator given by the left multiplication with x j ; note that X j p K ℓ q Ď K ℓ ` for all ℓ . By induction on the construction of q P R it is straightforwardto see that q p X q is well-defined for every q P R . Let ℓ “ t τ p r q , τ p s qu ´
2. ByProposition 3.6 there exist a finite-dimensional vector space U and a d -tuple of operators X on K ℓ ` ‘ U such that X P dom r X dom s and X j | K ℓ “ X j | K ℓ for j “ , . . . , d . A slight modification of Proposition 4.3 implies that r p X qr s “ r r s “ , s p X qr s “ r s s ‰ r q s P K denotes the image of q P V . (cid:3) The implication (i) ñ (ii) in Proposition 6.3 fails if only hermitian domains are con-sidered (e.g. take r “ x and s “ x ). It is also worth mentioning that while Proposition6.3 might look rather straightforward at first glance, there is a certain subtlety to it.Namely, the equivalence in Proposition 6.3 fails if only matrix tuples of a fixed size areconsidered. For example, let r “ x and s “ x x ; then dom r X dom s “ C andker r p X q Ď ker s p X q for all X P C , but dom p sr ´ q “ C zt u ˆ C (cf. [Vol17, Example2.1 and Theorem 3.10]). References [Bar02] A. Barvinok:
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Jurij Volˇciˇc, Department of Mathematics, Texas A&M University
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