aa r X i v : . [ m a t h . N T ] M a y Ax-Schanuel for Lie algebras
Georgios Papas
Abstract
In this paper we prove an Ax-Schanuel type result for the exponen-tial functions of Lie algebras over C . We prove the result first for thealgebra of upper triangular matrices and then for the algebra gl n ofall n × n matrices over C . We also obtain Ax-Lindemann type resultsfor these maps as a corollary, characterizing the bi-algebraic subsetsof these maps. § We study questions related to the functional transcedence of exponentialfunctions of n × n matrices over C . To be more precise, motivated by theexposition in [Pil15], we prove Ax-Schanuel and Ax-Lindemann type Theo-rems for the exponential function of upper triangular matrices, as well as theexponential function of general n × n matrices over C . We have divided ourexposition into two parts dealing with each of the cases separately.In the most general case we will consider the exponential function E : gl n → GL n over C . The strongest result that we achieve in this case is thefollowing Theorem (Full Ax-Schanuel for gl n ) . Let U ⊂ gl n be a bi-algebraic subvarietythat contains the origin, let X = E ( U ) , and let V ⊂ U and Z ⊂ X bealgebraic subvarieties, such that ~ ∈ V and I n ∈ Z . If C is a componentof V ∩ E − ( Z ) with ~ ∈ C , then, assuming that C is not contained in anyproper weakly special subvariety of U , dim C C ≤ dim C V + dim C Z − dim C X . Here the term component of a subset R ⊂ gl n refers to a complex-analytically irreducible component of R , while the term bi-algebraic refersto a subvariety U of gl n whose image E ( U ) under the exponential map is anopen subset of a subvariety of GL n . 1 short review of Ax-Schanuel and Ax-Lindemann Our main motivation is the Ax-Schanuel Theorem for the usual exponentialfunction of complex numbers. This result, originally a conjecture of Schanuel,is due to J. Ax [Ax71]. Ax’s proof employs techniques of differential algebra.One of the equivalent formulations of this theorem is the following
Theorem (Ax-Schanuel) . Let y , . . . , y n ∈ C [[ t , . . . , t m ]] have no constantterms. If the y i are Q − linearly independent then tr.d. C C ( y , . . . , y n , e y , . . . , e y n ) ≥ n + rank (cid:0) dy ν dt µ (cid:1) µ =1 ,...,mν =1 ,...,n . An immediate consequence of the above Ax-Schanuel Theorem is thecharacterization of all the bi-algebraic subsets of C n with respect to the map π : C n → ( C × ) n , given by ( z . . . , z n ) ( e z , . . . , e z n ). In other words itleads to a characterization of the subvarieties W ⊂ C n with the propertythat π ( W ) is an open subset of a subvariety of ( C × ) n . Definition.
A subvariety W of C n will be called weakly special, or geodesic,if it is defined by any number l ∈ N of equations of the form n X i = i q i,j z i = c j , j = 1 , . . . , l ,where q i,j ∈ Q and c j ∈ C . This characterization of bi-algebraic sets is due to the following result,dubbed Ax-Lindemann by Pila due to its resemblance to Lindemann’s theo-rem,
Theorem (Ax-Lindemann) . Let V ⊂ ( C × ) n be an algebraic subvariety. Thenany maximal algebraic algebraic subvariety W ⊂ π − ( V ) is weakly special. For more on these notions, along with a proof of Ax-Lindemann as acorollary of Ax-Schanuel, we refer to [Pil15].Subsequent results in functional transcendence that look to achieve sim-ilar results to the above theorem for other transcendental functions havealso been dubbed as ”Ax-Schanuel” and ”Ax-Lindemann” respectively. Ax-Schanuel results are known for affine abelian group varieties, due to J.Ax[Ax72], for semiabelian varieties, due to J. Kirby [Kir09], the j − function,due to J. Pila and J. Tsimerman [PT16], for Shimura varieties, due to N.Mok, J. Pila, and J. Tsimerman [MPT17], for variations of Hodge structures,due to B. Bakker and J. Tsimerman [BT17], and for mixed Shimura vari-eties due to Z. Gao [Gao18]. Finally, B. Klingler, E. Ullmo, and A. Yafaev[KUY16] have proven an Ax-Lindemann result for any Shimura variety.2 ummary of results in Part I The first part of this paper deals with the exponential of the algebra h n of n × n upper triangular elements. This map is more accessible to computations.These computations form the technical part of the reduction from the Ax-Schanuel result in this case to Theorem 2 . § E : h n → U n denote the exponential of h n , U n being the group ofupper triangular invertible matrices over C . Let A be an upper triangularmatrix with entries in C [[ t , . . . , t m ]]. We will denote the field extension of C that results from adjoining to C the entries of both matrices A and E ( A ) by C ( A, E ( A )). In this case our main result will be Theorem. (Ax-Schanuel for h n ) Let f , . . . , f n , g i,j ∈ C [[ t , . . . , t m ]] be powerseries, where ≤ i < j ≤ n . We assume that the f i don’t have a constantterm. Let A be the n × n upper triangular matrix with diagonal ~f and the ( i, j ) entry equal to g i,j . Let N = dim Q < f , . . . , f n > , then tr.deg C C ( A, E ( A )) ≥ N + rank( J ( ~f , ~g ; ~t )) . The main idea is that given a matrix A ∈ h n we are able to canonicallychoose a basis of generalized eigenvectors for it, based solely on the multi-plicities of its eigenvalues. This basis is chosen in such a way that makes itcomputable in terms of the entries of the matrix A . At the same time we candetermine the action of the matrix A in each of its generalized eigenspaces.The canonical basis and its properties lead us naturally to define thenotion of eigencoordinates in §
3. These will roughly be coordinates describingthe generalized eigenspaces of a matrix A along with the action of the matrix A in each of these spaces. Ultimately they will play the role of the g j inreducing the Ax-Schanuel result to Theorem 2 . Weakly special subvarieties for h n After establishing the Ax-Schanuel result we turn towards characterizing thebi-algebraic subvarieties of the exponential map E : h n → U n that containthe origin, i.e. the subvarieties W of h n that contain the origin and are suchthat E ( W ) is an open subset of a subvariety of U n . In the literature for othertranscendental maps such subvarieties are referred to as weakly special .To that end we start by defining the weakly special subvarieties of h n that contain the origin in §
5. Roughly speaking a subvariety V ⊂ h n thatcontains the origin will be weakly special if its diagonal coordinates satisfy Q − linear relations, while the other algebraic relations on it come from alge-braic relations on the eigencoordinates, i.e. from algebraic relations between3eneralized eigenvectors and the actions of matrices in V on their generalizedeigenspaces.As a corollary of our Ax-Schanuel we obtain an Ax-Lindemann-type re-sult. This result will imply that the weakly special sets we define will beexactly the bi-algebraic subsets of h n that contain the origin. We presenttwo examples of weakly special subvarieties as a motivation for the properdefinition later on in § Examples.
1. Let V ⊂ h be the set of all upper triangular × matrices A satisfying the following conditions:( C ) A has diagonal ( f , f , f ) wtih f i = f j for all i = j ,( C ) ~v = (1 , , is an f − eigenvector, and( C ) there exists s ∈ C such that ~v = ( s, s , is an f − eigenvector.Notice that all upper triangular matrices will have ~v = (1 , , as an eigen-vector for the eigenvalue f . We also note that here s can be thought of as afree variable.If we set W = Zcl ( V ) to be the Zariski closure of V in h n , then W willbe a weakly special subvariety of h n .2. Once again let us denote by ~v the vector (1 , , . We consider V ⊂ h n to be the set of all upper triangular matrices A satisfying the followingconditions:( C ) A has diagonal ( f , f , f ) wtih f = f ,( C ) let s ∈ C be such that ~v = ( s, , is an f − eigenvector,( C ) for the same s as above, the vector ~v = (0 , s , is a generalized eigen-vector for the eigenvalue f , and( C ) for the same s as above, we have that A~v = f ~v + s ~v .Here, as in the previous example, we can consider s as a free variable, in factit will correspond to one of the eigencoordinates.Finally, setting W = Zcl ( V ) we get a weakly special subvariety of h . ummary of results in Part II In the second part we deal with the same questions of functional transcen-dence this time for the exponential map E : gl n → GL n of the algebra of n × n matrices over C .In this case we generalize the picture we had in h n . Instead of a specificcanonical basis and eigencoordinates, we introduce the notion of the data of a matrix A ∈ gl n . This new notion will effectively have the role that theeigencoordinates had for h n .The data of a matrix A will comprise of the distinct eigenvalues of A ,their multiplicities, their generalized eigenspaces, and the nilpotent opera-tors defined by the matrix A on each such generalized eigenspace. Given thenumber k of distinct eigenvalues and the multiplicity m i , i = 1 , . . . , k , of eachof them, the rest of the above information, i.e. the generalized eigenspacesand corresponding nilpotent operators, will be parametrized by an affine vari-ety, which we will denote by W k ( ~m ). If we consider coordinates g j on W k ( ~m ),we shall see that the Ax-Schanuel result for E is reduced to Theorem 2 . A be an n × n matrix with entries in C [[ t , . . . , t m ]]. As before we willdenote the field extension of C that results from adjoining to C the entries ofthe matrices A and E ( A ) by C ( A, E ( A )). The result we obtain will then be Theorem (Ax-Schanuel for gl n ) . Let x i,j ∈ C [[ t , . . . , t m ]] be power serieswith no constant term, where ≤ i, j ≤ n , . Let z i ∈ C [[ t , . . . , t m ]] , where ≤ i ≤ n , denote the eigenvalues of the matrix A = ( x i,j ) . Let us also set N = dim Q < z , . . . , z n > , then tr.d. C C ( A, E ( A )) ≥ N + rank J (( x i,j ) | ~t ) . Weakly special subvarieties for gl n Again the above Ax-Schanuel result leads us to a description of the weaklyspecial subvarieties of gl n that contain the origin. These are defined in detailin §
7. Roughly speaking these will be subvarieties of gl n that are subject toalgebraic relations of the following two types:1. Q − linear relations on the eigenvalues and2. algebraic relations on the coordinates of a variety W k ( ~m ), as above, forsome k ∈ N and ~m ∈ N k , or in other words, relations coming from asubvariety W ⊂ W k ( ~m ).In other words, the relations are either on the eigenvalues, or between thegeneralized eigenspaces and the corresponding nilpotent operators defined on5hem. All the while there can be no algebraic relations between eigenvaluesand generalized eigenspaces or eigenvalues and nilpotent operators definedon those spaces. Alternatively, we require that the two types of relationsconsidered above do not interfere with one another.These results for the Lie algebra gl n will imply, as a corollary, Ax-Schanueland Ax-Lindemann type results for all subalgebras g of gl n and their respec-tive exponentials. Acknowledgements:
I would like to thank my advisor Jacob Tsimer-man for introducing me to the subject, for many helpful discussions, and forreading through earlier versions of this paper and making helpful suggestionsand pointing to some errors. I would also like to thank Edward Bierstone forhelpful remarks and useful discussions regarding the results of this paper. § In each of the two cases we deal with, we start by approaching the Ax-Schanuel result from a functional standpoint. We then use properties of theexponential maps in question to reduce to the following
Theorem 2.1.
Let f i , g j ∈ C [[ t , . . . , t m ]] , where ≤ i ≤ n , ≤ j ≤ k ,be power series. We assume that the f i don’t have a constant term. Then,assuming that the f i are Q − linearly independent modulo C , tr.deg C C ( { f i , g j , e f i : 1 ≤ i ≤ n, ≤ j ≤ k } ) ≥ n + rank( J ( ~f , ~g ; ~t )) . We note that Theorem 2 . . . § By the Seidenberg Embedding Theorem we may assume that the f i , g j areconvergent power series in some open neighbourhood B ⊂ C m . We considerthe uniformizing map π k : C n × C k → ( C × ) n , which is given by( x , . . . , x n , y , . . . , y k ) ( e x , . . . , e x n ). See [Sei58].
6e define D k = Γ( π k ), i.e. as a subset of C n × C k × ( C × ) n D k = { ( ~u, ~v ) : π k ( ~u ) = ~v } .Let us also consider ~G : B → D k to be the function defined by ~G ( t , . . . , t m ) = ( f , . . . , f n , g , . . . , g k , e f , . . . , e f n ),and finally we set U = ~G ( B ) and V = Zcl ( U ), the Zariski closure of U . Thendim C U = rank J ( ~f , ~g ; ~t )will be exactly the Jacobian that appears in the theorem we want to prove,while dim C ( V ) = tr.d. C C ( { f i , g j , e f i : 1 ≤ i ≤ n, ≤ j ≤ k } ). Furthermore,let π a be the projection on the first n coordinates of the space C n × C k × ( C × ) n ,and π m be the projection on the last n coordinates of the same space.The condition on the f i is then equivalent to the condition that the image π a ( U ) is not contained in the translate of a proper Q − linear subspace of C n or to the condition that π m ( U ) is not contained in the coset of a propersubtorus of ( C × ) n . Thus it suffices to prove Theorem 2.2 (Alternate Version 1) . Let D k and π k be as above and U ⊂ D k be an irreducible complex analytic subspace such that π m ( U ) is not containedin the coset of a proper subtorus of ( C × ) n . Then dim Zcl ( U ) ≥ n + dim C U . Again following [Tsi15] we can rephrase this last statement in terms ofsubvarieties of the space C n × C k × ( C × ) n . Theorem 2.3 (Alternate Version 2) . Let V ⊂ C n × C k × ( C × ) n be an irre-ducible algebraic subvariety, and U a connected complex-analytic irreduciblecomponent of V ∩ D k . Assuming that π m ( U ) is not contained in the coset ofa proper subtorus of ( C × ) n , then dim C V ≥ n + dim C U .Proof. We employ induction on k ≥
0. For k = 0 this is a consequence ofthe Ax-Schanuel Theorem .Assume that k ≥ k −
1. Then we considerthe projection p : C n × C k × ( C × ) n → C , See Theorem 1.3 in [Tsi15].
7f our space to the ( n + k ) − th coordinate, i.e. p ( x , . . . , x n , y , . . . , y k , z , . . . , z n ) = y k .Let also V = p ( V ) and, for y ∈ V , we consider the fibre V y of V over y . Similarly we consider the corresponding fibre U y of U over y . With thisnotation we get V = S y ∈ V { y } × V y .Since V is irreducible, if dim( V ) = 0 then V = { y } will be a single point.This implies that V = V ′ × { y } is isomorphic to an irreducible algebraicsubvariety V ′ ⊂ C n × C k − × ( C × ) n . In this case, U ⊂ V ∩ D k is isomorphicto a connected complex-analytic irreducible component of V ′ ∩ D k − and theresult follows by induction.We may therefore assume that dim V = 1. This tells us that V containsa non-empty affine open subset of C and that for y ∈ V generic we getdim V = dim V y + 1.The rest of the proof comprises of considering the only two possible casesfor the generic behaviour of the fibres U y . First Case:
Suppose that π a ( U y ) is generically not contained in thetranslate of a proper Q − linear subspace of C n .We have that V y ⊂ C n × C k − × ( C × ) n is an irreducible algebraic subvariety,and U y is a connected complex-analytic irreducible component of V y ∩ D k − .Therefore by the previous assumption and the inductive hypothesis we getthat for y ∈ V generic dim V y ≥ n + dim U y .This in turn implies that dim V ≥ n + (1 + dim U y ) and, since 1 + dim U y ≥ dim U , the result follows. Second Case:
If the assumption of the previous case doesn’t hold, thenfor y ∈ V chosen generically, π a ( U y ) ⊂ C n will be contained in the translate ofsome proper Q − linear subspace of C n . In other words U y ⊂ Z ( f y ), where f y = c ( y ) + n X i =1 q i ( y ) x i ∈ C [ x , . . . , x n ], is a linear polynomial with the coefficients q i ( y ) ∈ Q and c ( y ) ∈ C depending on y . At this point we consider another projection, namely we let Generically here refers to y ∈ V . Here Z ( f y ) = { ( x , . . . , x n ) : f y ( x , . . . , x n ) = 0 } , is just the set of solutions of f y = 0in C n . : C n × C k × ( C × ) n → C n × C k − × ( C × ) n be the projection given by p ( x , . . . , x n , y , . . . , y k , z , . . . , z n ) = ( x , . . . , x n , y , . . . , y k − , z , . . . , z n ).We also let V = p ( V ), U = p ( U ), V ′ = Zcl ( V ), the Zariski closure of V , and U ′ = ¯ U , the closure of U with respect to the standard topology on C n × C k − × ( C × ) n .For these new sets we get that V ′ is an irreducible subvariety of C n × C k − × ( C × ) n and U ′ is a connected irreducible complex-analytic componentof V ′ ∩ D k − . We also get that π a ( U ) = π a ( U ) and hence, by the initialassumption on U , π a ( U ′ ) is not contained in the translate of a Q − linearsubspace of C n . Therefore we may apply the inductive hypothesis to getdim V ′ ≥ n + dim U ′ .From the preceding discussion we get that dim V ≥ dim V = dim V ′ . Onthe other hand, since, by assumption, π a ( U ) is not contained in the translateof a Q − linear subspace of C n then the Z ( f y ), and hence the f y , will vary with y . This in turn implies that dim U = dim U = dim U ′ .Combining all of the above we reach the conclusion. § As a corollary of the above proof we are able to extract Ax-Schanuel resultsfor a larger family of spaces. The idea is that we are able to replace C k bya random affine variety. We approach this in a geometric setting similar tothe previous subsection.Let W be an affine variety over C and let π n : C n → ( C × ) n be the mapgiven by ( x , . . . , x n ) ( e x , . . . , e x n ).We consider the uniformizing map π n × id W : C n × W → ( C × ) n × W , theproduct of π n and the identity morphism id W of W . Let also p : ( C × ) n × W → ( C × ) n be the projection on ( C × ) n and let φ : C n × W → ( C × ) n be itscomposition with π n × id W .We also define D k ( W ) = Γ( φ ), i.e. as a subset of C n × W × ( C × ) n D k ( W ) = { ( ~u, ~v ) : φ ( ~u ) = ~v } . The coordinate function y k restricted to U will depend on the rest of the coordinatesof U . orollary 2.1. Let V ⊂ C n × W × ( C × ) n be an irreducible algebraic subvari-ety, and U a connected complex-analytic irreducible component of V ∩ D k ( W ) .Assuming that π m ( U ) is not contained in the coset of a proper subtorus of ( C × ) n , then dim C V ≥ n + dim C U .Proof. By Noether’s Normalization Lemma there exists a finite surjectivemorphism f : W → A d C where d = dim W . The product of this morphismwith the identity of C n × ( C × ) n in turn gives a finite morphism F : C n × W × ( C × ) n → C n × A d C × ( C × ) n .Indeed a morphism of affine varieties is finite if and only if it is proper , since F is proper as the product of two such morphisms it will also be finite. Theimage of the irreducible subvariety V under this map will be an irreduciblesubvariety V ′ of C n × A d C × ( C × ) n , since finite morphisms are closed.We also note that F maps the set D k ( W ) to the set D k ( A d C ). So thatthe closure U ′ = cl ( F ( U )) of the image of U with respect to the Euclideantopology will be a component of V ′ ∩ D k ( A d C ).Since F is finite we get that dim C U = dim C U ′ , dim C V = dim C V ′ , andby the construction of F it follows that π m ( U ′ ) is not contained in the cosetof a proper subtorus of ( C × ) n , since this is true for π m ( U ). Therefore theresult follows from Theorem 2 . Part I
Upper Triangular Matrices
Notation:
We will denote by U = U n the algebraic group of n × n uppertriangular invertible matrices over the field C , and by h = h n its Lie algebra,i.e. the algebra of n × n upper triangular matrices. Also we denote thecorresponding exponential map h n → U n by E = E n and its non-diagonalentries by E i,j , 1 ≤ i < j ≤ n .We will mainly concern ourselves with transcendence degrees over thefield C of extensions of the form C (Σ) with Σ a finite subset of some ring ofpower series, or a finite subset of regular functions on some variety over thefield C . Of particular interest will be the case were Σ is the set of entries ofa matrix A , or a matrix A and its exponential E ( A ). In that case we denote See Exercises II.4.1 and II.4.6 in [Har77]. C (Σ) over C by C ( A ) and C ( A, E ( A )) respectively.Our goal here is to state an Ax-Schanuel-type Theorem and reduce itsproof to Theorem 2 .
1. Our initial goal will be to find a lower bound for thetranscendence degree tr.d. C C ( A, E ( A )) = tr.d. C C ( { f i , g i,j , E i,j ( A ) , e f i : 1 ≤ i < j ≤ n } )where f i , g i,j ∈ C [[ t , . . . , t m ]], with 1 ≤ i < j ≤ n , and A = diag( ~f ) + ( g i,j ). § In this section we consider fixed f i , g i,j ∈ C [[ t , . . . , t m ]] without constantterms, with 1 ≤ i < j ≤ n , and let A = diag( ~f ) + ( g i,j ) be the matrix theydefine.The main idea is that eigenvectors and generalized eigenvectors for amatrix A will remain as such for the matrix E ( A ). As we will see shortlythe other information, that will naturally appear, and that we will have tokeep track of, are the nilpotent operators defined by A and E ( A ) on theirrespective generalized eigenspaces.In this section we define a canonically chosen basis of generalized eigen-vectors for a given matrix A ∈ h n that will only depend on the multiplicitiesof the eigenvalues of A . To this basis we can assign coordinates, which willbe rational functions on the entries of A . At the same time we achieve acanonical description of the respective nilpotent operators defined by A and E ( A ) on each of their generalized eigenspaces. To each such operator we willbe able to naturally assign certain rational functions of the entries of A . Thecombination of the above rational functions, both those describing the basisand those describing the nilpotent operators, will be what we will refer to as eigencoordinates .These new notions have a distinct advantage, as we will see, in our setting,when dealing with questions surrounding transcendence properties. Namelythey will allow us to:1. replace the g i,j by the eigencoordinates of A , when dealing with tran-scendence questions, and2. capture the essence of the map E , as far as transcendence is concerned,and replace the E i,j by the eigencoordinates of A , again in questions concern-ing transcendence. 11he scope of this section is to state and prove the main lemmas that wewill need concerning these new notions. As a motivation we first deal withthe case where all of the eigenvalues f i of our matrix A are distinct. Afterthat we proceed with dealing with the general case. § Let A , f i , and g i,j be as defined above, we assume that all the eigenvalues ofour matrix A are distinct. This is equivalent to the eigenvalues being distinctfor both A and E ( A ), since the f i have no constant term. Among all thepossible bases of eigenvectors for A we choose one in a canonical way.Let K be an algebraic closure of the field C ( { f i , g i,j : 1 ≤ i < j ≤ n } ),and let t i,j ∈ K , with 1 ≤ i < j ≤ n be such that the vector ~v i = ( − t ,i , . . . , − t i − ,i , , , . . . , f i − eigenvector for A . In this case ~v i will also be an e f i − eigenvectorfor E ( A ). We leave the proof of the existence of this canonical basis for A ,chosen as above, to Lemma 3 . Lemma 3.1.
Let A and t i,j be as above. Then tr.d. C C ( A ) = tr.d. C C ( { f i , t i,j : 1 ≤ i < j ≤ n } ) . Proof.
The condition
A~v i = ~v i translates to the following system of equations g i,j − g i,j − t j − ,j − . . . − g i,i +1 t i +1 ,j f i − f j = t i,j ... g j − ,j f j − − f j = t j − ,j (1)Since all of the f i are distinct we can write the t i,j as rational functions onthe entries of A by solving the above system of equations. In particular weget that for all i < j :a. t i,j ∈ Z [ f s − f j , g s,r : min( i, j ) ≤ s < r ≤ max( i, j )], andb. There exists Q i,j ∈ Z [ f s − f j , g s,r : min( i, j ) ≤ s < r ≤ max( i, j )] suchthat t i,j ( A ) = g i,j − Q i,j ( A ) f i − f j .Our result now follows trivially from the above remarks.12ince the matrices A and E ( A ) have the same eigenvectors, in the casewhere both A and E ( A ) have distinct eigenvalues, we get that, for all 1 ≤ i < j ≤ n , we will have t i,j ( A ) = t i,j ( E ( A )). This remark, together with theproof of Lemma 3 .
1, implies
Lemma 3.2.
Let A and t i,j be as defined above. Then tr.d. C C ( E ( A )) = tr.d. C C ( { e f i , t i,j ( A ) : 1 ≤ i < j ≤ n } ) , and tr.d. C C ( A, E ( A )) = tr.d. C C ( { f i , e f i , t i,j ( A ) : 1 ≤ i < j ≤ n } ) . Proof.
Let ~v i be the canonically chosen basis for A . Then, for all 1 ≤ i < j ≤ n , by combining the proof of Lemma 3 . t i,j ( A ) = t i,j ( E ( A )),there exists Q i,j as in the proof of Lemma 3 . E i,j ( A ) = ( g i,j − Q i,j ( A )) e fi − e fj f i − f j + Q i,j ( E ( A )).The result then follows trivially from these remarks. § In the case where we have eigenvalues with multiplicity greater than 1 wehave to alter our approach. The idea is to generalize the approach of theprevious subsection. In other words, we wish to find a canonically definedbasis, which will allow us to define coordinates that characterize our originalmatrix A uniquely. Furthermore we wish to describe those coordinates asalgebraic functions of the coordinates of A . Our ultimate goal is to obtainresults about transcendence degrees similar to those we proved in the previ-ous case.We assume that the matrix A has eigenvalues with multiplicities possiblygreater than 1. By our assumption that the f i have no constant term, eacheigenvalue e f i of E ( A ) has the same multiplicity as the respective eigenvalue f i of A . § Our first objective will be to describe and prove the existence of a certaincanonical basis for A . We begin by describing the canonical basis of eacheigenspace, then we combine these to create the basis we want.Just as before we let A = diag( ~f ) + ( g i,j ) an upper triangular matrix withentries in C [[ t , . . . , t m ]] and K be an algebraic closure of the field C ( { f i , g i,j :1 ≤ i < j ≤ n } ). 13 emma 3.3. Let z = f i = . . . = f i k − = f i k with i < . . . < i k . We alsoassume that f i = z for all i = i j . Let M ( z ) be the generalized eigenspace forthe eigenvalue z . Then there exists a unique basis B z of M ( z ) consisting ofvectors ~v i j , ≤ j ≤ k , such that1. ~v i j is of the form ~v i j = ( − t ,i j , . . . , − t i j − ,i j , , , . . . , , and2. t l,i j = 0 for l = i r where ≤ r ≤ j − , and3. there exist s i l ,i r ∈ K for ≤ l < r ≤ k such that A~v i j = z~v i j + s i ,i j ~v i + · · · + s i j − ,i j ~v i j − . Proof.
We proceed by induction on k = dim C ( M ( z )). For k = 1 the unique-ness follows from the unique solution to the t i,j described by the equations(1).Assume that k = 2. Then applying the system (1) we can determine thevector ~v i which will be an eigenvector for A . Since ~v i will in general be ageneralized eigenvector then we will have( A − zI n ) ~v i = s i ,i ~v i ,for some s i ,i ∈ K . We can therefore assume without loss of generality that t i ,i = 0.This relation will describe the coefficients of ~v i uniquely thanks to thefollowing series of equations:1. In the range i < j < i we get0 = − ( f i − − f i ) t i − ,i + g i − ,i ...0 = − ( f i +1 − f i ) t i +1 ,i − . . . − g i +1 ,i − t i − ,i + g i +1 ,i (2)2. For j = i − g i ,i +1 t i +1 ,i − . . . − g i ,i − t i − ,i + g i ,i = s i ,i (3)3. In the range 1 ≤ j < i − ( f j − f i ) t j,i − . . . − g j,i − t i − ,i + g j,i = s i ,i t j,i . (4)14olving the above system, starting from the first equation of (2) andmoving to the final equation described in the system (4) provides a uniquesolution for t j,i in terms of the f s and g s,t .Assume the result holds for dim C M ( z ) = k . Then in order to provethe inductive step we can create a similar system of equations with uniquesolution for the t i s ,i t in terms of the coefficient of the matrix A .We force relations on the canonical basis to be chosen so that A~v i j = z~v i j + s i ,i j ~v i + · · · + s i j − ,i j ~v i j − . (5)Then by induction it’s enough to determine the t s,i k +1 , since the rest ofthe vectors will constitute a basis for the respective eigenspace of a smallerdiagonal submatrix of A . To do this we just translate (5) for j = k + 1 to asystem of equations similar to the systems (2), (3), and (4).Combining all of the canonical bases of the eigenspaces we get a basis ~v i = ( − t ,i , . . . , − t i − ,i , , , . . . , ≤ i ≤ n , t i,j ∈ K for 1 ≤ i < j ≤ n , and such that t i,j = 0 if f i = f j .This will be the canonical basis of A . § In Lemma 3 . s i l ,i r for l < r . These coefficientsdetermine uniquely the nilpotent operator defined by A on the generalizedeigenspace M ( z ), i.e. they determine the nilpotent operator ( A − zI n ) | M ( z ) .In particular if z = f i = . . . = f i k then we have s i l ,i r ∈ K , where K is analgebraic closure of C ( A ) for 1 ≤ l < r ≤ k such that A~v i j = z~v i j + s i ,i j ~v i + · · · + s i j − ,i j ~v i j − . (6)From the proof of Lemma 3 . t i,j and the s i,j are rational func-tions of the f i and g i,j . Their combined information turns out to be exactlywhat we will need in what follows. Definition.
Let A = diag( ~f ) + ( g i,j ) be an upper triangular matrix withentries in C [[ t , . . . , t m ]] , such the diagonal entries f i have no constant term.Let ~v i = ( − t ,i , . . . , − t i − ,i , , , . . . , be a canonical basis for A . Also weconsider the s i,j that satisfy the equations in (6) for those eigenvalues of A with multiplicity greater than . We define the eigencoordinates of A tobe T i,j ( A ) = ( t i,j , if f i = f j s i,j , if f i = f j .
15t this point we want to replicate the results of Lemma 3 . . Lemma 3.4.
Let A be as above and let T i,j ( A ) be its eigencoordinates. Then tr.d. C C ( A ) = tr.d. C C ( { f i , T i,j ( A ) : 1 ≤ i < j ≤ n } ) . Proof.
For convenience we define the ring R i,j = Q [ g s,t , f s − f j : min( i, j ) ≤ s < t ≤ max( s, t ) , f s − f j = 0 ( s, t ) = ( i, j )].From the proof of Lemma 3 . T i,j are rational functionson the coordinates of the matrix such that(a) if f i = f j then there exists a Q i,j ∈ R i,j such that T i,j ( A ) = g i,j − Q i,j ( A ) f i − f j .(b) if f i = f j then there exists Q i,j ∈ R i,j such that T i,j ( A ) = g i,j − Q i,j ( A ).In other words the map A diag( f , . . . , f n ) + ( T i,j ( A )) is bijective and thecoordinates are rational functions on the entries of A with only factors of theform f i − f j appearing in the denominator.The equality of the transcendence degrees in question then follows easilyfrom the above remarks.The next step here is to study the effects of the exponential function onthe eigencoordinates. We record the main such results we will need in thefollowing Proposition 3.1.
Let A be as above and let T i,j ( A ) be its eigencoordinates.Then tr.d. C C ( E ( A )) = tr.d. C C ( { e f i , T i,j ( A ) : 1 ≤ i < j ≤ n } ) . From this and Lemma . we conclude that tr.d. C C ( A, E ( A )) = tr.d. C C ( { f i , e f i , T i,j ( A ) : 1 ≤ i < j ≤ n } ) . Proof.
We start by fixing some notation. If f i = f j we let R i,j be the ringsdefined in the proof of Lemma 3 .
4. Also, if f i = f j , we set D i,j = Q ( f j )[ f s − f j , g s,t : min( i, j ) ≤ s < t ≤ max( i, j ) , ( s, t ) = ( i, j )].16ince we have already dealt with the case where all eigenvalues are dis-tinct, we only have to study the behaviour of the exponential with respectto the generalized eigenspaces of dimension greater than 1.We assume that z = f i = . . . f i k with f i = z if i / ∈ { i , . . . , i k } so thatthe system (6) actually describes the eigencoordinates of A . We start bylooking at the effect of the exponential on the equations of (6).By induction and the definition of the exponential we get the followingrelation for the exponential matrix E ( A ) E ( A ) ~v i j = e z ~v i j + e z [ s i ,i j ~v i + · · · + s i j − ,i j ~v i j − ]++ e z [ S i ,i j ~v i + · · · + S i j − ,i j ~v i j − ] (7)where S i t ,i j = X t 4, and can therefore be considered as functions on A .Another consequence of (7) is that S i j − ,i j = 0 for all j , hence T i j − ,i j ( E ( A )) = e z ij T i j − ,i j ( A ) for all j . By induction on | t − l | , and the recursive definitionof the S i,j , it follows that we can write T i t ,i l ( A ) as a rational function on theentries of E ( A ) with one difference to the rational functions that have beenappearing so far, the denominator will be a product of factors of the form e f i − e f j , when f i = f j , and factors of the form e z , where z = f i = . . . = f i k .Indeed e − z s i j − ,i j ( E ( A )) = s i j − ,i j ( A ), so we get by (7) s i j − ,i j ( E ( A )) = e z s i j − ,i j ( A ) + e z s i j − ,i j − ( A ) s i j − ,i j ( A ).Using both of these together yields s i j − ,i j ( A ) = e − z ( s i j − ,i j ( E ( A ))) − e − z s i j − ,i j − ( E ( A )) s i j − ,i j ( E ( A )).The proof of this claim follows by induction along these lines.Combining these remarks with the proofs of Lemma 3 . . T i,j ( E ( A )) = e z ( T i,j ( A )) + P i,j ( E ( A )) , (9)where P i,j ∈ D i,j . 17e now go about proving that the transcendence degrees in question arein fact equal. From Lemma 3 . E ( A ) we get that tr.d. C C ( E ( A )) = tr.d. C C ( { e f i , T i,j ( E ( A )) : 1 ≤ i < j ≤ n } ) . From (9) on the other hand we get that tr.d. C C ( { e f i , T i,j ( A ) : 1 ≤ i < j ≤ n } ) ≤≤ tr.d. C C ( { e f i , T i,j ( E ( A )) : 1 ≤ i < j ≤ n } ) . We are therefore left with proving the converse inequality. We alreadyknow that T i,j ( E ( A )) = T i,j ( A ), if f i = f j . So suppose that as above we havesome repeating eigenvalues. By the definition of the S i,j in (7) and by (8)we conclude that all of the T i,j ( E ( A )) will in fact be elements of the field C ( { e f i , T i,j ( A ) : 1 ≤ i < j ≤ n } ). § h n At this point we are able to state and prove the Ax-Schanuel result for themap E : h n → U n . We also record a corollary of our result, as well as analternate geometric view in the spirit of [Pil15].As we have been doing so far, we let E i,j ( A ) be the corresponding entryof the matrix E ( A ) for 1 ≤ i < j ≤ n . Theorem 4.1. (Ax-Schanuel for h n ) Let f , . . . , f n , g i,j ∈ C [[ t , . . . , t m ]] bepower series, where ≤ i < j ≤ n . We assume that the f i don’t have aconstant term. Let A be the n × n upper triangular matrix with diagonal ~f and the ( i, j ) entry equal to g i,j . Then, assuming that the f i are Q − linearlyindependent modulo C , tr.deg C C ( A, E ( A )) ≥ n + rank( J ( ~f , ~g ; ~t )) .Proof. From Proposition 3 . tr.d. C C ( { f i , g i,j , e f i : 1 ≤ i < j ≤ n } ). This reduces the proofto Theorem 2 . 1, by giving the g i,j a new indexing g k , 1 ≤ k ≤ n ( n − .At this point we get the following immediate Corollary 4.1. Let f , . . . , f n , g i,j ∈ C [[ t , . . . , t m ]] be power series, where ≤ i < j ≤ n . We assume that the f i don’t have a constant term. Let A bethe n × n upper triangular matrix with diagonal ~f and the ( i, j ) entry equalto g i,j .Let N = dim Q < f , . . . , f n > , then r.deg C C ( A, E ( A )) ≥ N + rank( J ( ~f , ~g ; ~t )) . An Alternate Formulation In the spirit of [Pil15] we can give an alternate form of Theorem 4 . 1. Thistime the background is slightly changed. We let V ⊂ h n be an open subsetand X ⊂ V a complex analytic subvariety of V such that both X and V contain the origin and locally the coordinate functions, f , . . . , f n and g i,j for i < j , are meromorphic functions on X . We assume the same holds for thefunctions E i,j (diag( ~f ) + ( g s,t )), the functions e f i , and that there exist deriva-tions D i,j , 1 ≤ i ≤ j ≤ n , such that dim( X ) = rank( D i,j ( g s,t )), where herewe take g s,s = g s , and the rank is taken over the field of meromorphic func-tions on X . Finally we assume that the E s,t satisfy the expected differentialequations for D i,j .For reasons of convenience, and in keeping a similar notation to the pre-vious version, we let A = ( ~f ) + ( g s,t ) denote the matrix corresponding to thecoordinates f i and g i,j . Theorem 4.2. (Ax-Schanuel Alternate Formulation)In the above context, ifthe f i are Q − linearly independent modulo C , then tr.deg C C ( A, E ( A )) ≥ n + dim( X ) . Also similarly to above we can translate Corollary 4 . Corollary 4.2. In the above context, if the f i are are such that N = dim Q Let us restrict our attention to U Σ . On this dense open subset we can de-fine, by Lemma 3 . A ∈ U Σ . The eigen-coordinates of A will be well defined regular functions on U Σ , thanks againto the proof of Lemma 3 . . T i,j ( A ) willtranslate to an algebraic relation for the T i,j ( E ( A )) and vice versa. In order totranslate this into a geometric language we must first find a more convenientdescription for Z (Σ) and U Σ .We start by noting that we have an isomorphism Z (Σ) ∼ = L × A n ( n − C ,where L is a Q − linear subspace of C k where k is the number of genericallydistinct eigenvalues of Z (Σ). Let us also consider the dense open subset V Σ ⊂ C k with D Σ = C k \ ( [ ≤ i 4, which shows that we can change co-ordinates on the A n ( n − C part of the right hand side of the above isomorphismfrom g i,j to T i,j . In other words, we have an isomorphism T Σ : U Σ → L ′ × A n ( n − C , (11)where the A n ( n − C on the right signifies the affine space of the eigencoordinates T i,j . Conditions on the eigencoordinates Let V ⊂ A n ( n − C be an irreducible subvariety of A n ( n − C that contains theorigin, where the latter is considered as the space of the eigencoordinates.Then if we consider W = L ′ × V this will be an irreducible subvariety of L ′ × A n ( n − C . We now consider its inverse under the isomorphism T Σ of (11).Finally we consider the Zariski closure of the resulting set in h n , which wewill denote by X (Σ , V ).Notice that X (Σ , V ) satisfies exactly what we wanted, the diagonal coor-dinates are only subject to Q − linear equations, any relation on the strictlyupper triangular part comes from relations on the eigencoordinates, it isirreducible and it contains the origin. Definition (Weakly Special Subvarieties) . An irreducible subvariety X of h n that contains the origin will be called weakly special if there exist:1. a system Σ of Q − linear equations on the diagonal entries, and2. an irreducible subvariety V defined as above,such that X = X (Σ , V ) , where the latter is as defined in the above discussion. § h n and other corollaries Here we record some corollaries of our Ax-Schanuel result. We start witha corollary about atypical intersections and then use that to prove the Ax-21indemann result. The latter allows us to characterize the bi-algebraic sub-sets for the map E that contain the origin. The exposition follows in thespirit of [Pil15].We start by defining the notion of a component. Definition. Let W ⊂ h n and V ⊂ U n be algebraic subvarieties. Then a component X of W ∩ E − ( V ) will be a complex-analytically irreducible com-ponent of W ∩ E − ( V ) . The context in which we will be using our Ax-Schanuel result is the onedescribed in Theorem 4 . Theorem 6.1 (Full Ax-Schanuel for h n ) . Let U ⊂ h n be a weakly specialsubvariety, containing the origin, and set X = E ( U ) . Let W ⊂ U and V ⊂ X be algebraic subvarieties, with n ∈ W and I n ∈ V . If the component C of W ∩ E − ( V ) that contains the origin is not contained in any properweakly special subvariety of U then dim C C ≤ dim C V + dim C W − dim C X .Proof. Following the discussion of the previous section, we can associate tothe subvariety U a system Σ of Q − linear equations on the diagonal entries, aswell as the corresponding Q − linear subspace L of C k where k is the numberof generically distinct eigenvalues, and a subvariety Z of A n ( n − C . In otherwords with the notation of the previous section U = X (Σ , Z ). We also denoteby U Σ the corresponding dense open subset we had in the discussion of theprevious section.At this point we let B = U Σ ∩ C , then B is again a complex analyticallyirreducible subset that is dense in C and it is not contained in a properweakly special subvariety of U . In particular we will have dim C B = dim C C .We denote by f i the diagonal coordinates of a matrix as functions on B and similarly for the coordinates g i,j . Likewise we denote the diagonalcoordinates of the exponential map by E i and the strictly upper triangularby E i,j and we consider them as functions on B as well, keeping in mind that E i ( A ) = e f i . For reasons of convenience we let A = diag( ~f ) + ( g i,j ) denotethe matrix of the corresponding coordinates.We start with some simple remarks concerning our setting. First of all,we will have dim C W ≥ tr.d. C C ( A ), and (12)dim C V ≥ tr.d. C C ( E ( A )) . (13)22ext, we employ Corollary 4 . 2, to get that, if N = dim Q < f , . . . , f n > ,then dim C B + N ≤ tr.d. C C ( A, E ( A )) . (14)We also set m = tr.d. C C ( A, E ( A )), and l = tr.d. C C ( { T i,j ( A ) : 1 ≤ i < j ≤ n } ),with T i,j denoting once again the eigencoordinates of a matrix. From thispoint on for convenience we will denote simply by T i,j the elements T i,j ( A ).At this point we turn our attention to Lemma 3 . 4, Proposition 3 . 1, alongwith equations (8) and (9). On B the eigencoordinates T i,j are well definedas functions on B . From the aforementioned lemmas we also get tr.d. C C ( A ) = tr.d. C C ( { T i,j , f i : 1 ≤ i < j ≤ n } ) , and (15) tr.d. C C ( E ( A )) = tr.d. C C ( { T i,j , E i : 1 ≤ i < j ≤ n } ) . (16)By the definition of weakly special subvarieties we see thatdim C U = dim C E ( U ) = dim C X = N + l. (17)On the other hand, by the minimality of the weakly special subvariety U , weget that, if K = C ( { T i,j : 1 ≤ i < j ≤ n } ), m = l + tr.d. K K ( { E i , f i : 1 ≤ i ≤ n } )We also have that tr.d. C C ( A ) = tr.d. C K + tr.d. K K ( f , . . . , f n ),and likewise that tr.d. C C ( E ( A )) = tr.d. C K + tr.d. K K ( E , . . . , E n ).Combining the above equalities implies that m ≤ tr.d. C C ( E ( A )) + tr.d. C C ( A ) − l. (18)Using (18) along with (13) and (12) yields m ≤ dim C W + dim C V − l .Together with (14), (17), and the fact that dim C B = dim C A , this finishesthe proof. 23 orollary 6.1 (Ax-Lindemann for h n ) . Let V ⊂ U n be an algebraic subva-riety with I n ∈ V . If W ⊂ E − ( V ) is a maximal subvariety that contains theorigin, then W is a weakly special subvariety.Proof. Let U be the minimal weakly special subvariety that contains W , X = E ( U ), and let V ′ = V ∩ X . We use Theorem 6 . A = W to getdim C W ≤ dim C W + dim C V ′ − dim C X .This implies dim C X ≤ dim C V ′ , and since V ′ ⊂ X we get that X ⊂ V andthat W ⊂ U ⊂ E − ( V ). Maximality of W then implies that W = U isweakly special. Part II General Matrices Having studied the exponential of h n we can expect to achieve similar Ax-Schanuel and Ax-Lindemann results for the case of general matrices. Onceagain the key role will be played by the eigenvalues of our matrix.We start with considering certain subsets of gl n that will assist us in for-mulating the Ax-Schanuel and Ax-Lindemann results. We then proceed ina similar fashion to the upper triangular case. Namely we start by statingthe Ax-Schanuel result and then reduce its proof to Theorem 2 . 1. Finally,we conclude with some corollaries of our result. Notation: For the remainder we will denote the Lie algebra of n × n matrices over C by gl n and the respective exponential function by E : gl n → GL n . § gl n We begin our study by defining the data of a matrix A , a notion that willgeneralize the eigencoordinates we had in the upper triangular case. Withthe help of this new notion we can define, as we will see, the weakly specialsubvarieties and achieve a simpler description of the exponential.As we did in the case of the upper triangular matrices, throughout thissection we present as lemmas the equalities of transcendence degrees that wewill need in the proofs of our main results.24 Let V be a C − linear space with dim C V = n . Let also A ∈ Hom( V, V ) = gl n then A is uniquely characterized by the following data:1. A number of distinct complex numbers z , . . . , z k , the eigenvalues of A ,2. for each eigenvalue z i an m i ∈ N , the multiplicity of that eigenvalue,such that k X i =1 m i = n ,3. for each z i as above, a subspace V i ≤ V , with dim V i = m i , suchthat V ∼ = k M i =1 V i , i.e. to every eigenvalue a corresponding generalizedeigenspace, and4. for each z i as above, a nilpotent operator N i ∈ Hom( V i , V i ), i.e. N i =( A − z i I n ) | V i .The above picture also holds over an arbitrary algebraically closed field. Definition. Let A be a matrix as above. Then we define the data of thematrix to be the data { ( z , . . . , z k ) , ( m , . . . , m k ) , ( V , . . . , V k ) , ( N , . . . , N k ) } . The information of the generalized eigenspaces V i and nilpotent operators N i of a matrix A with k distinct eigenvalues is parametrized by a varietywhich we will denote by W k ( ~m ). We also let w k, ~m = dim W k ( ~m ). In whatfollows we will need to consider a set of coordinates on such a variety, whichwe will denote by g j with 1 ≤ j ≤ w k, ~m .What’s most important in our setting is that this passage from a matrixto its data preserves the transcendence degree. As in the upper triangularcase, we start by considering elements in some ring of formal power series.In this setting we indeed have Lemma 7.1. Let x i,j ∈ C [[ t , . . . , t l ]] , ≤ i, j ≤ n . We assume that the x i,j have no constant term and that the matrix A = ( x i,j ) has exactly k distincteigenvalues z , . . . , z k with respective multiplicities m i . Let K be an algebraicclosure of the field C ( A ) and g j , ≤ j ≤ w k, ~m , be coordinates for the variety W k ( ~m ) parametrizing the corresponding data of A . Then tr.d. C C ( A ) = tr.d. C C ( { z i , g j : 1 ≤ i ≤ k, ≤ j ≤ w k, ~m } ) . roof. Follows in a similar fashion to Lemma 3 . 4. The procedure of changingfrom either set of functions to the other is algebraic. This whole process ispossible since we are working over the algebraically closed field K .More precisely, the z i and g j are algebraic functions of the x i,j determinedby the process of finding the Jordan canonical form. On the other handstarting with the z i and the g j we can determine the Jordan canonical formof a matrix and therefore also its entries as algebraic functions of the z i and g j . In particular we get that C ( { z i , g j : 1 ≤ i ≤ k, ≤ j ≤ w k, ~m } ) is analgebraic extension of the field C ( A ). § Let A = ( x i,j ) be a matrix with x i,j ∈ C [[ t , . . . , t l ]], where the x i,j have noconstant term. Let us also assume that A has data given by { ( z , . . . , z k ) , ( m , . . . , m k ) , ( V , . . . , V k ) , ( N , . . . , N k ) } .We are able to consider such data working over an algebraic closure K of thefield C ( A ). We would like to extract from this a simpler way for computingthe effect of the exponential on A .Since the x i,j , and hence also the z i , have no constant term, it’s easy tosee that the corresponding data for the matrix E ( A ) will be:1. the eigenvalues will be the distinct elements e z , . . . , e z k ∈ C [[ t , . . . , t l ]],2. the multiplicities m i will be the same,3. the generalized eigenspaces V i will remain as such, and4. the nilpotent operator corresponding to each e z i is N ′ i = e z i ( E ( N i ) − id V i ).Once again the same will hold if we substitute the field C by any algebraicallyclosed field.Let E i,j ( A ) denote the ( i, j ) entry of the exponential matrix E ( A ). Thenwe will have the following Proposition 7.1. Let x i,j ∈ C [[ t , . . . , t l ]] , ≤ i, j ≤ n , be such that the x i,j have no constant term. We assume that A = ( x i,j ) has exactly k distincteigenvalues z , . . . , z k with respective multiplicities m , . . . , m k . Let K bean algebraic closure of the field C ( A ) and g j = g j ( A ) , ≤ j ≤ w k, ~m , be oordinates for the variety W k ( ~m ) parametrizing the corresponding data of A . Then tr.d. C C ( E ( A )) = tr.d. C C ( { e z i , g j : 1 ≤ i ≤ k, ≤ j ≤ w k, ~m } ) , and tr.d. C C ( A, E ( A )) = tr.d. C C ( { z i , e z i , g j : 1 ≤ i ≤ k, ≤ j ≤ w k, ~m } ) . Proof. Let h j = g j ( E ( A )), 1 ≤ j ≤ w k, ~m , be the coordinates in W k ( ~m )parametrizing the corresponding data of E ( A ). From Lemma 7 . E ( A ) we get tr.d. C C ( E ( A )) = tr.d. C C ( { e z i , h j : 1 ≤ i ≤ k, ≤ j ≤ w k, ~m } ) . Therefore we are left with proving the following equality tr.d. C C ( { e z i , h j : 1 ≤ i ≤ k, ≤ j ≤ w k, ~m } ) == tr.d. C C ( { e z i , g j : 1 ≤ i ≤ k, ≤ j ≤ w k, ~m } ) . By the remarks above though the g j and h j will parametrize the same V i ,so that their only difference is located in those h j and g j that parametrizethe nilpotent operators. For the latter we know that we will have N ′ i = e z i ( E ( N i ) − id V i ). Claim: The map f : N il ( m ) → N il ( m ) given by N E ( N ) − id is abialgebraic map, where N il ( m ) denotes the space of nilpotent operators onan m − dimensional C − vector space.Assuming this claim holds, if h ij , g ij , j ∈ J i denote the elements amongthe h j , and g j respectively, that parametrize the information of the nilpotentoperators N ′ i and N i , then the above shows that C ( { e z i , g ij : j ∈ J i } ) = C ( { e z i , h ij : j ∈ J i } ),for all i = 1 , . . . , k . Combining this with the fact that h j = g j for all of therest, i.e. those parametrizing the V i , the result follows trivially.The above argument shows that in fact C ( { e z i , g j : 1 ≤ j ≤ w k, ~m } ) = C ( { e z i , h j : 1 ≤ j ≤ w k, ~m } ). Combining this with the remark at the endof the proof of Lemma 7 . F = C ( { A, e z i , g j : 1 ≤ i ≤ k, ≤ j ≤ w k, ~m } ) is a finite algebraic extension of the field C ( A, E ( A )).Similarly, F = C ( { e z i , g j : 1 ≤ i ≤ k, ≤ j ≤ w k, ~m } ) is a finite algebraicextension of F which finishes the proof of the second equality.27 roof of Claim. Let N be a nilpotent operator on an m − dimensional vectorspace and let k ∈ N be such that N k = 0 and N k +1 = 0. Then E ( N ) = k X r =0 N r r ! so that N E ( N ) − id is obviously algebraic.On the other hand define L : N il ( m ) → N il ( m ) given by N log( id + N ) = ∞ X r =1 ( − r +1 N r r .Since our operator are nilpotent this sum is finite and the map is algebraic,similarly to the above argument. The two functions are inverse of each other,which proves the claim. § We return once more to gl n and proceed towards defining the weakly specialsubvarieties. The results we had so far lead us in a natural way to considersome specific subsets of gl n . We start by formalizing some of the previousideas, mainly that of the variety W k ( ~m ).Consider a vector space V over C with dim C V = n , some fixed k ∈{ , . . . , n } , some fixed m i ∈ N for 1 ≤ i ≤ k such that m + . . . + m k = n .Then W k ( ~m ) is the algebraic variety over C that parametrizes the informa-tion { ( V , . . . , V k ) , ( N , . . . , N k ) } , where V i ≤ V are subspaces that definea decomposition of V such that dim C V i = m i and each N i is a nilpotentoperator on the respective subspace V i .At this point we let X k ( ~m ) = A k C × W k ( ~m ). It is therefore natural toconsider coordinates z i , 1 ≤ i ≤ k , for A k and coordinates g j , 1 ≤ j ≤ w k, ~m for W k ( ~m ), as we did before. We also define the mapΦ k, ~m : X k ( ~m ) → gl n such that { ( z i ) , ( g j ) } 7→ P J P − , with J being a block diagonal matrix, withits blocks being Jordan blocks, where we allow elements of the super diagonalto be either 0 or 1, and P being the transition matrix that is defined by thesubspaces V i parametrized by the coordinates g j . Remarks. 1. The image of Φ k, ~m will contain all matrices with k distincteigenvalues each of them having multiplicity m i . This is true since the Jordancanonical form of a matrix is uniquely determined up to permutation by theJordan blocks. Permuting these blocks also results in respective permutationsof the columns of the transition matrix P , which are parametrized by the g j .2. We note that Φ k, ~m is not injective, but is a quasi-finite morphism ofvarieties. This suffices for our purposes. elations on Eigenvalues We expect that the only algebraic relations that will be allowable on theeigenvalues will be Q − linear relations. Since we have already accounted forthe number of distinct eigenvalues we also require that these relations don’tforce any more eigevalues to be equal.With that in mind we let Σ be a finite system of equations of the form f ( ~z ) = k X i =1 q i z i = 0 on the distinct z i , and let I be the ideal generated by Σ in C [ z , . . . , z k ]. We also assume that ∄ i, j such that z i − z j ∈ I , i.e. the systemof equations Σ doesn’t force any of the previously distinct z i to coincide.Finally we also let T Σ = Z (Σ) ⊂ A k . Other Relations For the rest of the data of the matrix we allow any algebraic relationthat does not depend on the eigenvalues. So we consider W ⊂ W k ( ~m ) tobe a subvariety of W k ( ~m ). Then if we are given a system Σ as above and asubvariety W ⊂ W k ( ~m ) we let X ( k, ~m, Σ , W ) = T Σ × W .All of the above lead us naturally to the following definition. Definition. A subvariety U ⊂ gl n containing the origin will be called weaklyspecial if there exist a natural number ≤ k ≤ n , a vector ~m = ( m , . . . , m k ) ∈ N k such that k X i =1 m i = n , a system Σ of Q − linear equations, and a subvariety W ⊂ W k ( ~m ) , all defined as above, such that U = Zcl (Φ k, ~m ( X ( k, ~m, Σ , W )) ,where Zcl ( R ) denotes the Zariski closure in gl n of a subset R ⊂ gl n . § gl n We continue with our study of E : gl n → GL n by stating the Ax-Schanuelresult and proving it by reducing to Theorem 2 . E by E i,j . We start withstating the theorem in the functional point of view.29 heorem 8.1 (Ax-Schanuel for gl n ) . Let x i,j ∈ C [[ t , . . . , t m ]] be power serieswith no constant term, where ≤ i, j ≤ n , . Let z i ∈ C [[ t , . . . , t m ]] , where ≤ i ≤ n , denote the eigenvalues of the matrix A = ( x i,j ) . Let us also set N = dim Q < z , . . . , z n > , then tr.d. C C ( A, E ( A )) ≥ N + rank J (( x i,j ) | ~t ) .Proof. Let K be the algebraic closure of the field C ( A ) and assume A =( x i,j ) ∈ gl n ( K ) has exactly k distinct eigenvalues. Let us also assume thatthe data of the matrix A is given by( z , . . . , z k ),( m , . . . , m k ), ( V , . . . , V k ), and ( N , . . . , N k ).Let { g j : 1 ≤ j ≤ w k, ~m } be coordinates in W k ( ~m )( K ) describing the abovedata.We are therefore in a position to apply Proposition 7 . tr.d. C C ( A, E ( A )) = tr.d. C C ( { z i , e z i , g j : 1 ≤ i ≤ k, ≤ j ≤ w k, ~m } ),which reduces to Theorem 2 . An Alternate Formulation Similar to the alternate formulation Theorem 4 . h n we can give an alternate form of Theorem 8 . 1, again we have tochange the background accordingly.We let V ⊂ gl n be an open subset and X ⊂ V a complex analyticsubvariety of V containing the origin such that locally the functions x i,j for1 ≤ i, j ≤ n , are meromorphic functions on X . We assume the same holds forthe functions E i,j (( x s,t )) and that there exist derivations D i,j , 1 ≤ i ≤ j ≤ n ,such that dim C X = rank( D i,j ( x s,t )), where the rank is taken over the fieldof meromorphic functions on X . Finally we assume that the E s,t satisfy theexpected differential equations for D i,j .Once again, for reasons of convenience, and notational coherence, we let A = ( x i,j ) denote the matrix corresponding to the coordinates x i,j . Theorem 8.2. (Ax-Schanuel Alternate Formulation) In the above context,if the eigenvalues z i are such that N = dim Q < z , . . . , z n > , then tr.deg C C ( A, E ( A )) ≥ N + dim C X . This version of the Ax-Schanuel result is the one most useful when ex-tracting geometric corollaries, as we have already seen.30 gl n and other corollar-ies We approach this in the same way as we did for the corresponding result in § 6. We start with defining components in this setting. After that we provea result on atypical intersections, similar to Theorem 6 . 1, and then, just asin § 6, use this to infer our Ax-Lindemann result. Definition. Let W ⊂ gl n and V ⊂ GL n be algebraic subvarieties. Thena component C of W ∩ E − ( V ) will be a complex-analytically irreduciblecomponent of W ∩ E − ( V ) . Theorem 9.1 (Full Ax-Schanuel for gl n ) . Let U ⊂ gl n be a weakly specialsubvariety that contains the origin, let X = E ( U ) , and let V ⊂ U and Z ⊂ X be algebraic subvarieties, such that ~ ∈ V and I n ∈ Z . If C is a componentof V ∩ E − ( Z ) with ~ ∈ C , then, assuming that C is not contained in anyproper weakly special subvariety of U , dim C C ≤ dim C V + dim C Z − dim C X .Proof. Let U = Zcl (Φ k, ~m ( X ( k, ~m, Σ , W ))), consider the set B k = X ( k, ~m, Σ , W ) \{ z i − z j ∈ πi Z : 1 ≤ i, j ≤ k } ,and let B = Φ k, ~m ( B k ) ∩ C . We will have dim C B = dim C C and on B the x i,j , z i , and g j are well defined meromorphic functions, where z i and g j denote the coordinates of T Σ and W ⊂ W k ( ~m ) respectively. We also let N = dim Q < z , . . . , z k > .Then we may use Theorem 8 . tr.d. C C ( A, E ( A )) ≥ N + dim C C. (19)On the other hand, we have the following inequalities:dim C Z ≥ tr.d. C C ( E ( A )) , anddim C V ≥ tr.d. C C ( A ) . Combining these with Lemma 7 . . C Z ≥ tr.d. C C ( { e z i , g j : 1 ≤ i ≤ k, ≤ j ≤ w k, ~m } ) , and (20)dim C V ≥ tr.d. C C ( { z i , g j : 1 ≤ i ≤ k, ≤ j ≤ w k, ~m } ) . (21)31n the other hand, if we set L = C ( { g j : 1 ≤ j ≤ w k, ~m } ), and let M = tr.d. C L , we get that dim C X = N + M. (22)By the minimality of U , in containing C , and hence B , we also get that tr.d. C C ( { z i , g j : 1 ≤ i ≤ k, ≤ j ≤ w k, ~m } ) == M + tr.d. L L ( { z i : 1 ≤ i ≤ k } ) , (23) tr.d. C C ( { e z i , g j : 1 ≤ i ≤ k, ≤ j ≤ w k, ~m } ) == M + tr.d. L L ( { e z i : 1 ≤ i ≤ k } ) , and (24) tr.d. C C ( A, E ( A )) = M + tr.d. L L ( { z i , e z i : 1 ≤ i ≤ k } ) . (25)Combining these with (20) and (21) we get thatdim C V ≥ M + tr.d. L L ( { z i : 1 ≤ i ≤ k } ), anddim C Z ≥ M + tr.d. L L ( { e z i : 1 ≤ i ≤ k } ).The rest of the proof follows similar to that of Theorem 6 . § 6, we conclude with the characterization of bi-algebraic setsthat contain the origin. Corollary 9.1 (Ax-Lindemann for gl n ) . Let Z ⊂ GL n be an algebraic sub-variety with I n ∈ Z . If V ⊂ E − ( Z ) is a maximal subvariety that containsthe origin, then V is a weakly special subvariety.Proof. Let U be the minimal weakly special subvariety that contains V , X = E ( U ) and let Z ′ = Z ∩ E ( U ). We use Theorem 9 . C = V to getdim C V ≤ dim C V + dim C Z ′ − dim C X .This implies dim C X ≤ dim C Z ′ , and since Z ′ ⊂ X we get that X ⊂ Z andthat V ⊂ U ⊂ E − ( Z ). Maximality of V then implies that V = U is weaklyspecial. Remarks. 1. We note that the above results imply, as a direct corollary,Ax-Schanuel and therefore also Ax-Lindemann results for all Lie subalgebrasof gl n .2. In mimicking the classical Ax-Schanuel statement, we can extract Ax-Schanuel and Ax-Lindemann type statements for Cartesian powers of theexponential map of a Lie algebra from our results.Even more generally, we can infer such results for the Cartesian productsof exponentials E i : g i → G i of Lie algebras g i , ≤ i ≤ r . We achieve thisby noticing that the exponential of the Lie algebra g = g × . . . × g r is theCartesian product of the E i . eferences [Ax71] James Ax. On Schanuel’s conjectures. Ann. of Math. (2) , 93:252–268, 1971.[Ax72] James Ax. Some topics in differential algebraic geometry. I. Ana-lytic subgroups of algebraic groups. Amer. J. Math. , 94:1195–1204,1972.[BT17] Benjamin Bakker and Jacob Tsimerman. The ax–schanuel conjec-ture for variations of hodge structures. 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Ax-Schanuel for the j -function. Duke Math. J. , 165(13):2587–2605, 2016.[Sei58] Abraham Seidenberg. Abstract differential algebra and the analyticcase. Proceedings of the American Mathematical Society , 9(1):159–164, 1958. 33Tsi15] Jacob Tsimerman. Ax-Schanuel and o-minimality. In O-minimalityand diophantine geometry , volume 421 of London Math. Soc. Lec-ture Note Ser. , pages 216–221. Cambridge Univ. Press, Cambridge,2015. Dept. of Mathematics, University of Toronto, Toronto, Canada. E-mail address : [email protected]@mail.utoronto.ca