Bilinear systems with two supports: Koszul resultant matrices, eigenvalues, and eigenvectors
Matías Bender, Jean-Charles Faugère, Angelos Mantzaflaris, Elias Tsigaridas
aa r X i v : . [ c s . S C ] M a y Bilinear systems with two supports:Koszul resultant matrices,eigenvalues, and eigenvectors
Mat´ıas R. Bender , Jean-Charles Faug`ere ,Angelos Mantzaflaris , and Elias Tsigaridas Sorbonne Universit´e,
CNRS , INRIA ,Laboratoire d’Informatique de Paris 6,
LIP6 ,´Equipe
PolSys ,4 place Jussieu, F-75005, Paris, France Johann Radon Institute for Computational and AppliedMathematics (RICAM),Austrian Academy of Sciences, Linz, AustriaMarch 2018 ∗ Abstract
A fundamental problem in computational algebraic geometry isthe computation of the resultant. A central question is when andhow to compute it as the determinant of a matrix. whose elementsare the coefficients of the input polynomials up-to sign. This prob-lem is well understood for unmixed multihomogeneous systems, that ∗ This work is based on a paper originally published in ISSAC ’18, July 16–19, 2018,New York, USA [BFMT18] s for systems consisting of multihomogeneous polynomials with thesame support. However, little is known for mixed systems, that is forsystems consisting of polynomials with different supports.We consider the computation of the multihomogeneous resultantof bilinear systems involving two different supports. We present aconstructive approach that expresses the resultant as the exact de-terminant of a Koszul resultant matrix , that is a matrix constructedfrom maps in the Koszul complex. We exploit the resultant matrix topropose an algorithm to solve such systems. In the process we extendthe classical eigenvalues and eigenvectors criterion to a more generalsetting. Our extension of the eigenvalues criterion applies to a generalclass of matrices, including the Sylvester-type and the Koszul-typeones.
Keywords:
Resultant; Sparse Resultant; Determinantal formula; Bilinearsystem; Mixed Multihomogeneous system; Polynomial solving
The resultant is a central object in elimination theory and computationalalgebraic geometry. We use it to decide when an overdetermined polynomialsystem has a solution and to solve well-defined (square) systems. Moreover,it is one of the few tools that take into account the sparsity of supports ofthe polynomials.Usually, we compute the resultant as a quotient of determinants of twomatrices [Mac02, Jou97, DD01, D’A02]. If we can compute the resultant asa determinant of only one matrix whose non-zero entries are forms evaluatedat the coefficients of the input polynomials, then we have a determinantalformula . Among these cases, the best we can hope for is to have linear forms.In general, determinantal formulas do not exist and it is an open problem todecide when they do.The matrices appearing in the computation of resultants have a strongstructure and we can classify them according to it. For a system ( f , . . . , f n ),a Sylvester-type formula is a matrix that represents a map ( g , . . . , g n ) P i g i f i . It extends the classical Sylvester matrix and it corresponds to thelast map of the Koszul complex of ( f , . . . , f n ). Another kind of formula isthe Koszul-type formula that involves the other maps of the Koszul com-plex. We call the matrices related to this formula
Koszul resultant matri- es [MT17, BMT17]. For both formulas, the elements of the matrices arelinear polynomials in the coefficients of ( f , . . . , f n ). Other important resul-tant matrices include B´ezout- and
Dixon-type ; we refer to [EM99] and ref-erences therein for details. We consider
Koszul-type determinantal formulas for mixed multihomogeneous bilinear systems with two supports.A well-known tool to derive determinantal formulas [WZ94, DE03, EM12,EMT16, MT17, BMT17] is the Weyman complex [Wey94], a generalizationof the Koszul complex. For an introduction we refer to [Wey03, Sec. 9.2] and[GKZ08, Sec. 2.5.C, Sec. 3.4.E]. We follow this approach.For unmixed multihomogeneous systems, that is systems where all thepolynomial share the same support, determinantal formulas are well studied,e.g., [SZ94, WZ94, KS97, CK00, DD01, Wey03, DE03]. On the other hand,when we consider polynomials with different supports, that is mixed systems ,little is known about determinantal formulas; with the exception of scaledmultihomogeneous systems [EM12], that is when the supports are scaledcopies of one of them, and the bivariate tensor-product case [MT17, BMT17].The resultant is also a tool to solve 0-dimensional square polynomialsystems ( f , . . . , f n ). There are different variants, for example by hiding avariable, or using the u-resultant; we refer to [CLO06, Chp. 3] for a generalintroduction. When a Sylvester-type formula is available, we can use thecorresponding resultant matrix to obtain the matrix of the multiplicationmap of a polynomial f in K [ x ] / h f , . . . , f n i . Then, we can solve the systemby computing the eigenvalues and eigenvectors of the latter matrix, e.g.,[AS88, Emi96]. The eigenvalues correspond to the evaluation of f at everyzero of the system. From the eigenvectors we can recover the coordinates ofthe zeros. To our knowledge similar techniques involving matrices comingfrom Koszul-type formulas do not exist up to now.We consider mixed bilinear polynomial systems. On the one hand, this issimplest case of mixed multihomogeneous systems where no resultant formulawas known. On the other hand, bilinear, and their generalization multilinear,polynomial systems are common in applications, for example in cryptography[FLDVP08, Jou14] and game theory [EV14]. We refer to [FSEDS11], see also[Spa12], for computing the roots of unmixed multilinear systems by means ofGr¨obner bases, and to [EMT16] by using resultants. We refer to [BFT18] fora Gr¨obner bases approach to solve square mixed multihomogeneous systems.3 ur contribution We introduce a new algorithm to solve square mixedmultihomogeneous systems consisting of bilinear polynomials with two differ-ent supports. It relies on eigenvalues and eigenvectors computations. Follow-ing classic resultant techniques we add a polynomial, f , to make the systemoverdetermined. The polynomial f must be trilinear, as this is simplest onethat can separate the roots. Then, we introduce a determinantal formulafor the resultant of this overdetermined system. This is the first determi-nantal formula for a mixed multilinear polynomial system. Using Weyman’scomplex, we derive a Koszul-type formula and compute the resultant as thedeterminant of a
Koszul resultant matrix .We present a general extension of the eigenvalue criterion that works fora general class of formulas (see Def. 4.1), which include the Koszul-type andSylvester-type formulas as special cases. We consider a square matrix M whose determinant is a multiple of the resultant of a system ( f , . . . , f n ). Ifthere is a monomial x σ in f such that we can partition M as (cid:2) M , M , M , M , (cid:3) where M , is invertible, the coefficient of the monomial x σ in f appearssolely in the diagonal of M , and this diagonal contains only this coefficient,then the evaluations of f ( x ) x σ at the solutions of ( f , . . . , f n ), that is { f ( x ) x σ | x = α :( ∀ i > f i ( α ) = 0 , x σ | x = α = 0 } , are eigenvalues of the Schur complement of M , , that is M , − M , · M − , · M , .We extend the eigenvector criteria for these mixed bilinear systems. When M is our Koszul resultant matrix , we show how to recover the coordinates ofthe solutions from the eigenvectors of the Schur complement of M , . Thisapproach works for systems whose solutions have no multiplicities.Algorithm 1 summarizes our strategy to solve square 0-dimensional 2-bilinear systems whose solutions have no multiplicities. Future work.
Weyman complex leads to determinantal formulas for mixedmultihomogeneous systems. A possible extension is to classify all the pos-sible determinantal formulas for mixed multihomogeneous systems of thisconstruction, similarly to [WZ94]. The structure of the Koszul resultantmatrix could lead to more efficient algorithms to perform linear algebrawith these matrices, and hence to solve faster, theoretically and practically,square mixed multihomogeneous systems. Finally, our eigenvector criterionshould be extensible to any Koszul resultant matrix. This approach mightbe adapted to recover the coordinates of the solutions with multiplicities.4 lgorithm 1
Solve2Bilinear (( ¯ f , . . . , ¯ f n )) Input: ( ¯ f , . . . , ¯ f k ) is a square 2-bilinearsystem such that V P ( ¯ f , . . . , ¯ f k ) isfinite and has no multiplicities. A ← Random linear change of coordinates preserving the structure. ( f , . . . , f n ) ← ( ¯ f ◦ A, . . . , ¯ f n ◦ A ). (Thm. 4.7) f ← Random trilinear polynomial in S (1 , , (cid:2) M , M , M , M , (cid:3) ← (cid:26) Matrix corresponding to δ (( f , . . . , f n ) , m ), split wrtthe monomial w θ . (Def. 4.1) (cid:8)(cid:0) f w θ ( α ) , ¯ v α (cid:1)(cid:9) α ← (cid:26) Set of pairs Eigenvalue-Eigenvector of the Schurcomplement of M , . (Thm. 4.2) for all (cid:0) f w θ ( α ) , ¯ v α (cid:1) ∈ (cid:8)(cid:0) f w θ ( α ) , ¯ v α (cid:1)(cid:9) α do Extract the coordinates α x , α y from ρ α ( b λ α ) by recovering itfrom (cid:2) M − , · M , I (cid:3) · ¯ v . (Thm. 4.13) Let α z ∈ P n z be the unique solution to the linear system given by { f ( α x , α y , z ) = 0 , . . . , f n ( α x , α y , z ) = 0 } , over K [ z ]. Recover the solution of the system ( ¯ f , . . . , ¯ f n ), as A (cid:0) ( α x , α y , α z ) (cid:1) . end forPaper organization In Sec. 2 we introduce notation and the resultantof mixed multihomogeneous systems. In Sec. 3, we present the Weymancomplex in our setting and we prove the existence of a Koszul-type formula.Then, in Sec. 4, we present algorithms for solving 2-bilinear systems; Sec. 4.1extends the eigenvalue criterion to a general class of matrices and Sec. 4.2studies the eigenvectors to recover the coordinates of the solutions. Finally,in Sec. 5, we compare the size of our matrix with the experimental size ofthe matrices in Gr¨obner basis computation.
Consider n x , n y , n z ∈ N and let P := P n x × P n y × P n z be a multiprojec-tive space over an algebraic closed field K of characteristic 0. Consider x := { x , . . . , x n x } , y := { y , . . . , y n y } , z := { z , . . . , z n z } and let S x ( d x ) := K [ x ] d x , S y ( d y ) := K [ y ] d y , and S z ( d z ) := K [ z ] d z be the spaces of homo-geneous polynomials in variables x , y and z and degrees d x , d y and d z ,respectively. Let S ( d x , d y , d z ) := S x ( d x ) ⊗ S y ( d y ) ⊗ S z ( d z ) be the multi-5omogeneous polynomials in x , y , and z of degrees d x , d y , and d z , re-spectively. We say that the polynomials in S ( d x , d y , d z ) have multidegree d := ( d x , d y , d z ) ∈ N . To avoid the repetition of the various definitions for x , y , and z , we consider t ∈ { x, y, z } . The dual space of S t ( d t ) is S t ( d t ) ∗ .For σ t ∈ N n t +10 , we define t σ t := Q n t i =0 t σ t,i i . Then A ( d t ) := { σ t : t σ t ∈ S t ( d t ) } is the set of the exponents of all the monomials of degree d t in t and A ( d ) := A ( d x ) × A ( d y ) × A ( d z ) is the set of all the exponents of the monomi-als of multidegree d . If σ = ( σ x , σ y , σ z ) ∈ A ( d ), then w σ := x σ x y σ y z σ z . Let n := n x + n y + n z . For multidegrees d = ( d , . . . , d n ) ∈ ( N ) n +1 , we considersquare multihomogeneous polynomial system f := ( f , . . . , f n ) ∈ S ( d ) × · · · × S ( d n ) . (1)Let V P ( f ) be the set of solutions of f over P . The multihomogeneous B´ezoutbound ( MHB ) [VdW78] bounds the number of isolated solutions of f over P [Ber75, Kus76, Kho78]. The bound is attained for any generic squaresystem f . It is the mixed volume of the polytopes A ( d ) , . . . , A ( d n ) [CLO06,Chp. 7] and appears as the coefficient of the monomial Q t ∈{ x,y,z } X n t t in Q nj =1 P t ∈{ x,y,z } d j,t X t [MS87].In the sequel we consider overdetermined systems which we construct byadding an f ∈ S ( d ) to f , that is, f := ( f , f , . . . , f n ) ∈ S ( d ) × · · · × S ( d n ) . (2)Typically, we will consider d = (1 , , f to be as simpleas possible while still depending on all the variables. The multihomogeneous sparse resultant of f is a polynomial in the co-efficients of the polynomials in f , which vanishes if and only if the sys-tem has a solution over P . Following [CLO06], for fixed d . . . d n ∈ N ,we introduce a set of variables u i := { u i, σ } σ ∈A ( d i ) , for 0 ≤ i ≤ n , and u := { u , . . . , u n } . Given P ∈ K [ u ], we let P ( f ) denote the value ob-tained by replacing each variable u i, σ with the coefficient of the mono-mial w σ in the polynomial f i of f . In this way we obtain polynomi-als over the coefficients of a polynomial system. The “universal” system6 d ,...,d n ∈ K [ u ][ x , y , z ] × · · · × K [ u n ][ x , y , z ] is F d ,...,d n := (cid:16) X σ ∈A ( d ) u , σ w σ , . . . , X σ ∈A ( d n ) u n, σ w σ (cid:17) . (3)Here the variables of u parametrize the systems described by polynomials in S ( d ) × · · · × S ( d n ) over K A ( d ) × · · · × K A ( d n ) .Consider the set of all tuples of n + 1 multihomogeneous polynomialstogether with their common solutions over P , { ( f , . . . , f n , α ) ∈ S ( d ) × · · · × S ( d n ) × P : ( ∀ ≤ i ≤ n ) f i ( α ) = 0 } . The projection of this set on S ( d ) ×· · ·× S ( d n ) is the set of overdetermined systems with common solutions in P , { ( f , . . . , f n ) ∈ S ( d ) × · · · × S ( d n ) : V P ( f , . . . , f n ) = ∅} . By the ProjectiveExtension Theorem [CLO92, Chp. 8 Sec. 5], this projection is a closed setunder the Zariski topology and it forms an irreducible hypersurface over thevector space S ( d ) × · · · × S ( d n ) [GKZ08, Chp. 8]. More formally, there is anirreducible polynomial Res P ( d , . . . , d n ) ∈ Z [ u ] such that for all the systems f ∈ S ( d ) × · · · × S ( d n ), V P ( f ) = ∅ if and only if Res P ( d , . . . , d n )( f ) = 0.This polynomial is the sparse resultant over P for multihomogeneous systemsof multidegrees ( d , . . . , d n ).The resultant Res P ( d , . . . , d n ) is itself a multihomogeneous polynomial,homogeneous in each block of variables u i . For each i , its degree with respectto u i is MHB ( d , . . . , d i − , d i +1 , . . . , d n ). A square of type ( n x , n y , n z ; r, s ) is a bilinear system f := ( f , . . . , f n ) with two different supports, namely f , . . . , f r ∈ S (1 , , f r +1 , . . . , f n ∈ S (1 , , n = r + s , n y ≤ r and n z ≤ s . Itholds MHB ( f ) = (cid:0) rn y (cid:1)(cid:0) n − rn z (cid:1) . Example 2.1.
The following (Eq. (4) ) is a square 2-bilinear system of type (1 , , , and has two solutions over P , namely α := (1 : 1 ; 1 : 1 ; 1 : 1) and α := (1 : 3 ; 1 : 2 ; 1 : 3) . f := 7 x y − x y − x y + 2 x y f := − x y + 7 x y − x y − x y f := − x z + 9 x z − x z − x z . (4)7onsider the trilinear f ∈ S (1 , , f :=( f , f , . . . , f n ) as overdetermined 2-bilinear systems. We can also consider f in S (1 , , S (1 , , S (1 , , S (0 , ,
0) and S (0 , , f because in the other cases it is not always possible to separateall the solutions of V P ( f ). Example 2.2 (Cont.) . Consider the overdetermined 2-bilinear system f :=( f , f , f , f ) , where f := 3 x y z − x y z − x y z + 2 x y z + x y z + 2 x y z + 2 x y z − x y z . In the following, we use F (2) to denote the “universal” system of overde-termined 2-bilinear systems (see Sec. 2.1). Similarly, we use Res (2) P , for theresultant of the “universal” system F (2) . Lemma 2.3.
Let
MHB ( f ) = (cid:0) rn y (cid:1)(cid:0) sn z (cid:1) . The degree of Res (2) P is µ := ( n x + 1) MHB ( f ) r · s − n y · n z + r + s + 1( r − n y + 1)( s − n z + 1) . (5) A complex K • is a sequence of modules { K v } v ∈ Z together with homomor-phisms δ v : K v → K v − , such that ( ∀ v ∈ Z ) Im( δ v ) ⊆ Ker( δ v − ), i.e., δ v ◦ δ v − = 0. We say that the complex is exact if ( ∀ v ∈ Z ) Im( δ v ) =Ker( δ v − ). A complex is bounded when there are two constants a and b such that for every v < a or b < v , it holds K v = 0. If all the K v are finitedimensional free-modules, then we can choose a basis of them and we can rep-resent the maps δ v using matrices. Under certain assumptions (see [GKZ08,App. A]) given a bounded complex of finite dimensional free-modules we candefine its determinant. It is the quotient of minors of the matrices of δ v andit is not zero if and only if the complex is exact. If there are only two non-zero modules of the same dimensions in the complex (that is all the othermodules are the zero module), the determinant of the complex reduces tothe determinant of the (matrix of the) map between these modules.The Weyman Complex [Wey94, WZ94, Wey03] of a multihomogeneoussystem f is a bounded complex that is exact if and only if the sparse resultant8f the system f does not vanish [Wey03, Thm. 9.1.2]. The determinant ofthe complex is a power of the resultant [Wey03, Prop. 9.1.3]. When all themultidegrees are bigger than zero, the determinant of this complex is a non-zero constant multiple of the sparse resultant [GKZ08, Thm. 3.4.11]. If theWeyman Complex only involves two non-zero modules, the resultant of thecorresponding system is the determinant of the map between these modules,and it has a determinantal formula.Let f := ( f , f , . . . , f n ) be an overdetermined 2-bilinear system. Con-sider E := K n +1 and its canonical basis e , . . . , e n . Given a set I ⊂ { , . . . , n } ,we define e I := e I ∧ · · · ∧ e I I as the exterior product of the elements e I , . . . , e I I . As the exterior product is antisymmetric, that is e i ∧ e j = − e j ∧ e i , when we write e I ∧ · · · ∧ e I I we assume that ( ∀ i ) I i < I i +1 . Let V a,b,c E be the vector space over K generated by { e K ∪ I ∪ J : K ⊂ { } , I ⊂{ , . . . , r } , J ⊂ { r + 1 , . . . , n } , I = a, J = b, K = c } .For a degree vector m ∈ Z , the Weyman complex is K • ( f , m ). Eachmodule of the complex is K v ( m ) := L n +1 p =0 K v,p ( m ), where K v,p ( m ) := M a + b + c = p ≤ a ≤ r ≤ b ≤ s ≤ c ≤ H p − v P ( m − ( p, p − b, p − a )) ⊗ ^ a,b,c E, and H q P ( m ′ ) is the q -th cohomology of P with coefficients in the sheaf O ( m ′ ),and the space of global sections is H P ( m ′ ) [Har77]. Note that the terms K v,p ( m ) do not depend on f [WZ94, Prop. 2.1]. Since P is a product ofprojective spaces, by K¨unneth’s formula H p − v P (cid:0) m ′ x , m ′ y , m ′ z (cid:1) ∼ = O t ∈{ x,y,z } H j t P nt ( m ′ t ) , (6)where j x + j y + j z = p − v . By Serre’s duality [Har77, Ch.III,Thm. 5.1] wehave the identifications: Proposition 3.1.
For each t ∈ { x, y, z } , m ′ t ∈ Z , it holds(1) H P nt ( m ′ t ) ∼ = S t ( m ′ t ) if m ′ t ≥ , (2) H n t P nt ( m ′ t ) ∼ = S t ( − m ′ t − − n t ) ∗ if m ′ t < n t , where “ ∗ ” denotes the dual space, and (3) H q P nt ( m ′ t ) ∼ = 0 , of allother values of q and m t . As a corollary from Eq. (6), for each t ∈ { x, y, z } , j t ∈ { , n t } . Moreover,we can identify dual complexes. 9 roposition 3.2 ([Wey03, Thm. 5.1.4]) . Let m and m ′ be degree vectorssuch that m + m ′ = ( n y + n z , n x + n z − s, n x + n y − r ) . Then, K v ( m ) ∼ = K − v ( m ′ ) ∗ for all v ∈ Z and K • ( f , m ) is dual to K • ( f , m ′ ) . If K ( m ), K ( m ) are the only non-zero modules in the Weyman complex K • ( f , m ), then the determinant of the complex is the determinant of themap, between them, δ ( f , m ). In this case, we have a determinantal formulafor the resultant. In the following, when it is clear from the context, we write δ instead of δ ( f , m ). Theorem 3.3.
Let f be a 2-bilinear overdetermined system of type ( n x , n y , n z ; r, s ) , with f ∈ S (1 , , . The degree vectors(1) ( n y − , − , n x + n y − r + 1) , (2) ( n z + 1 , n x + n z − s + 1 , − ,(3) ( n z − , n x + n z − s + 1 , − , (4) ( n y + 1 , − , n x + n y − r + 1) lead to determinantal Weyman complexes for Res (2) P ( f ) . Observation 3.4.
The four degree vectors of Thm. 3.3 provide a single ma-trix formula. Vector 1 (resp. 2) is obtained from 3 (resp. 4) by exchangingthe variables y and z . By Prop. 3.2, we can see that 1,2 and 3, 4 are dualpairs, yielding the same matrix transposed.Proof. We consider only the first degree vector m :=( n y − , − , n x + n y − r + 1). By Obs.3.4, the other cases are similar.First, we show that the complex has only two non-zero terms. Since K v ( m ) := L n +1 p =0 K v,p ( m ), and in view of Eq. (6), for each K v,p ( m ), we haveto consider sums P t ∈{ x,y,z } j t = p − v . By Prop. 3.1, if j t
6∈ { , n t } , then K v,p = 0. The remaining cases are summarized in the following table andtheir analysis follows. j x j y j z Case0 n x j x j y j z Case0 n z (1) n x n z (1) j x j y j z Case0 n y n z (2) n x n y n z (2) j x j y j z Case n y n x n y (4) Case 1: j y = 0. The second term in the tensor product of K v,p is H P ny ( − − a − c ) ∼ = S y ( − − a − c ), by Prop. 3.1. As a, c ≥ S y ( − − a − c ) = 0.Hence, K v,p = 0. 10 ase 2: j z = n z . The third term in the tensor product of K v,p is H n z P nz ( n x + n y − r + 1 − b − c ) ∼ = S z ( − ( n x + n y + n z ) + r − b + c ) ∗ , by Prop. 3.1. As n x + n y + n z = r + s , − ( n x + n y + n z ) + r − b + c = − s − b + c < b ≤ s and c ≤
1. Hence, H n z P nz ( n x + n y − r + 1 − b − c ) = 0 and so K v,p = 0. Case 3: j x = 0, j y = n y . As j y = n y , the second term in the tensorproduct K v,p is H n y P ny ( − − a − c ) ∼ = S y ( a + c − n y ) ∗ , by Prop. 3.1. Thismodule is not zero iff a + c ≥ n y . Consider the first term in the tensorproduct, H P nx ( n y − − p ) ∼ = S x ( n y − − p ). If a + c ≥ n y , as p = a + b + c ,then n y − − p ≤ − − b <
0. Hence, either H n y P ny ( − − a − c ) = 0 or H P nx ( n y − − p ) = 0, and so K v,p = 0. Case 4: j x = 0, j y = n y , j z = 0. The first term in the tensor product K v,p is H n x P nx ( n y − − p ) ∼ = S x ( − n x − n y + p ) = S x ( v ), as p − v = j x + j y + j z = n x + n y .Hence, H n x P nx ( n y − − p ) = 0 iff v ≥
0. As j z = 0 the third term in the tensorproduct of K v,p is H P nz ( n x + n y − r + 1 − b − c ) ∼ = S z ( n x + n y − r + 1 − b − c ).This term is not zero iff n x + n y − r + 1 ≥ b + c . Moreover, as p = a + b + c , v = a + b + c − n x − n y . Then, if H P nz ( n x + n y − r + 1 − b − c ) = 0, then v ≤ a − r + 1. By definition a ≤ r , so v ≤ K ,n x + n y +1 ( m ) and K ,n x + n y ( m ) are equal to zero. Hence, by [Wey03, Prop. 9.1.3] the deter-minant of (a matrix expressing) δ is a power of Res (2) P ( f ).To conclude, it suffices to show that the exponent is equal to one. Due tothe form δ : K ,q +1 ( m ) → K ,q ( m ), the elements in a matrix that represents δ have degree ( q + 1) − q = 1 as polynomials in K [ u ] [Wey03, Prop. 5.2.4].Therefore, the exponent is one iff the degree of the resultant is equal to thedimension of the matrix of K • ( f , m ) : 0 → K ,n x + n y +1 ( m ) δ −→ K ,n x + n y ( m ) → . We analyze the possible values for ( a, b, c ) to compute the dimension. Fol-lowing
Case 4 , if H P nz ( n x + n y − r + 1 − b − c ) = 0, then the possible valuesfor a are v + r − ≤ a ≤ r , for v ∈ { , } . As b = p − a − c , and 0 ≤ c ≤ a, b, c ) and write our modules as The exponent is known to be one for any very ample supports [GKZ08], i.e.( ∀ i, j ) d i,j >
0. However, due to the zero degrees, 2-bilinear supports are ample butnot very ample. = K ,n x + n y +1 ∼ = L , ⊕ L , (7)= (cid:16) S x (1) ∗ ⊗ S y ( r − n y ) ∗ ⊗ S z (0) ⊗ ^ r,s − n z +1 , E (cid:17) ⊕ (cid:16) S x (1) ∗ ⊗ S y ( r − n y + 1) ∗ ⊗ S z (0) ⊗ ^ r,s − n z , E (cid:17) .K = K ,n x + n y ∼ = L , ⊕ L , ⊕ L , ⊕ L , (8)= (cid:16) S x (0) ∗ ⊗ S y ( r − n y − ∗ ⊗ S z (0) ⊗ ^ r − ,s − n z +1 , E (cid:17) ⊕ (cid:16) S x (0) ∗ ⊗ S y ( r − n y ) ∗ ⊗ S z (1) ⊗ ^ r,s − n z , E (cid:17) ⊕ (cid:16) S x (0) ∗ ⊗ S y ( r − n y ) ∗ ⊗ S z (0) ⊗ ^ r − ,s − n z , E (cid:17) ⊕ (cid:16) S x (0) ∗ ⊗ S y ( r − n y + 1) ∗ ⊗ S z (1) ⊗ ^ r,s − n z − , E (cid:17) . To compute their dimensions we notice that dim (cid:16)V a,b,c E (cid:17) = (cid:0) ra (cid:1)(cid:0) sb (cid:1) ,and we recall that dim S t ( q ) = dim S t ( q ) ∗ = (cid:0) n t + qq (cid:1) . The calculation leads todim( K ) = dim( K ) = µ , see Eq. (5).The four degree vectors of Thm. 3.3 are not the only ones that lead todeterminantal formulas. We are interested in them because, experimentally,there are no Sylvester-type formulas and only these degree vectors lead toKoszul-type formulas [EMT16, MT17]. δ ( f , m ) Following [Wey03, Sec. 5.5], we construct the map δ ( f , m ) : K ( m ) → K ( m ). By Obs. 3.4, we only consider m = ( n y − , − , n x + n y − r + 1).In the proof of Thm. 3.3 we saw that the map δ ( F (2) , m ) has linearcoefficients in K [ u ]. As it is a linear map between free modules, it is enoughto define it over a basis of K and K .First we introduce some notation. Let t ∈ { x, y, z } . For each σ t ∈ A ( d ), d ∈ N , consider ∂t σ t ∈ S t ( d ) ∗ such that ∂t σ t ( P c θ t t θ t ) = c σ t . The set12 ∂t σ t : σ t ∈ A ( d ) } forms a basis of S t ( d ) ∗ . The map ⋆ t : K [ t ] × K [ t ] ∗ → K [ t ] ∗ ,acts as ( t θ t , ∂t σ t ) t θ t ⋆ t ∂t σ t , where t θ t ⋆ t ∂t σ t = ∂t σ t − θ t if ( ∀ i, ≤ i ≤ n t ) σ t,i ≥ θ t,i . (9)This map is graded, that is, for each ( d, ¯ d ) ∈ Z , it maps the elements in S t ( d ) × S t ( ¯ d ) ∗ to S t ( ¯ d − d ) ∗ . We will denote the map by “ ⋆ ” when the variableis clear from the context. We define the graded map ψ , ψ : ( K [ x ] ∗ ⊗ K [ y ] ∗ ⊗ K [ z ]) × ( K [ x ] ⊗ K [ y ] ⊗ K [ z ]) → ( K [ x ] ∗ ⊗ K [ y ] ∗ ⊗ K [ z ]) (10) ψ ( ∂x σ x ⊗ ∂y σ y ⊗ z σ z , x θ x ⊗ y θ y ⊗ z θ z ) :=( x θ x ⋆ ∂x σ x ) ⊗ ( y θ y ⋆ ∂y σ y ) ⊗ ( z θ z + σ z )For each ( d x , d y , d z , ¯ d x , ¯ d y , ¯ d z ) ∈ Z , it maps ( S x ( d x ) ∗ ⊗ S y ( d y ) ∗ ⊗ S z ( d z )) × (cid:0) S x ( ¯ d x ) ∗ ⊗ S y ( ¯ d y ) ∗ ⊗ S z ( ¯ d z ) (cid:1) to S x ( d x − ¯ d x ) ∗ ⊗ S y ( d y − ¯ d y ) ∗ ⊗ S z ( d z + ¯ d z ).As δ ( f , m ) : K → K is linear and K ∼ = L , ⊕ L , , we define the mapover a basis of L , and L , . For each ℓ ∈ S x (1) ∗ ⊗ S y ( r − n y ) ∗ ⊗ S z (0) and e I ∈ V r,s − n z +1 , E , we consider ℓ ⊗ e I ∈ L , and δ ( f , m ) ( ℓ ⊗ e I ) := n x + n y +1 X i =1 ( − i − ψ ( ℓ , f I i ) ⊗ e I \{ I i } ∈ L , ⊕ L , . For each ℓ ∈ S x (1) ∗ ⊗ S y ( r − n y + 1) ∗ ⊗ S z (0) and e J ∈ V r,s − n z , E , we consider ℓ ⊗ e J ∈ L , and δ ( f , m ) ( ℓ ⊗ e J ) := n x + n y +1 X i =1 ( − i − ψ ( ℓ , f J i ) ⊗ e J \{ J i } ∈ L , ⊕ L , ⊕ L , . The map δ ( f , m ) corresponds to a Koszul-type formula, involving mul-tiplication and dual multiplication maps. The matrix that represents thismap is a Koszul resultant matrix [MT17, BMT17].13 xample 3.5 (Cont.) . In this case, m = (0 , − , . We consider the fol-lowing monomial basis,Basis of K (Columns)(A) ∂x ∂y e { , , } (B) ∂x ∂y e { , , } (C) ∂x ∂y e { , , } (D) ∂x ∂y e { , , } (E) ∂x ∂y e { , , } (F) ∂x ∂y e { , , } (G) ∂x ∂y e { , , } (H) ∂x ∂y ∂y e { , , } (I) ∂x ∂y e { , , } (J) ∂x ∂y ∂y e { , , } Basis of K (Rows)(I) e { , } (II) e { , } (III) ∂y e { , } (IV) ∂y e { , } (V) ∂y e { , } (VI) ∂y e { , } (VII) ∂y z e { , } (VIII) ∂y z e { , } (IX) ∂y z e { , } (X) ∂y z e { , } The following matrix represents δ ( f , m ) wrt the basis above. ( A ) ( B ) ( C ) ( D ) ( E ) ( F ) ( G ) ( H ) ( I ) ( J )( I ) 0 0 0 − II ) 0 0 0 − − III ) 0 − − − ( IV ) − − − ( V ) 0 − − ( V I ) − − ( V II ) 0 − − − ( V III ) − − − ( IX ) 0 − − − ( X ) − − − he × splitting illustrated above will be used in the next section. Consider a 0-dimensional system f , . . . , f n ∈ K [ x ]. A common strategyfor solving is to work over K [ x ] / h f , . . . , f n i , which is a finite a dimensionalvector space over K . We fix a monomial basis, choose f ∈ K [ x ], and com-pute the matrix that represents the multiplication by f in the quotient ring.Its eigenvalues are the evaluations of f at the solutions. For a suitablebasis, from the eigenvectors we can recover the coordinates of all the solu-tions [EM07, CLO06, Cox05]. To compute these matrices we can use theSylvester-type formulas [AS88, Emi96, CLO06]. We extend these techniquesto a general family of matrices, that includes the Koszul resultant matrix(Sec. 3.2). In this section we assume fixed multidegrees d , . . . , d n . Definition 4.1 (property Π θ ) . Given θ ∈ A ( d ) and a matrix M := (cid:2) M , M , M , M , (cid:3) ∈ K [ u ] K×K (Sec. 2.1), we say that M has the property Π θ ( d , . . . , d n ) , or simply Π θ , when: • Res P ( d , . . . , d n ) divides det( M ) , • the submatrix M , is square and its diagonal entries equal to u , θ , and • the coefficient u , θ does not appear anywhere in M expect from thediagonal of M , . For a system f , Eq. (2), let M ( f ) be the specialization of M at f (see Sec. 2.1). If M , ( f ) is invertible, then the Schur complement of M , ( f )is M , ( f ) − M , ( f ) · ( M , ( f )) − · M , ( f ). To simplify, we write ( M , − M , · M − , · M , )( f ). Theorem 4.2.
Consider θ ∈ A ( d ) and a matrix M ∈ K [ u ] K×K such that Π θ holds (Def. 4.1). Assume a system f , Eq. (2) , such that the specialization M , ( f ) is non-singular. Then, for all α ∈ V P ( f ) such that w θ ( α ) = 0 , f w θ ( α ) is an eigenvalue of the Schur complement of M , ( f ) . roof. The idea of the proof is as follows: For each α ∈ V P ( f ), Eq. (1),we consider a system g , slightly different from f , with α as a solution.We study the matrices M ( f ) and M ( g ) and from the kernel of M ( g ) weconstruct an eigenvector for the Schur complement of M , ( f ) correspondingto an eigenvalue equal to f w θ ( α ).Let α ∈ V P ( f ) such that w θ ( α ) = 0. Consider the polynomial g := f − f w θ ( α ) · w θ and a new system g := ( g , f , . . . , f n ). The coefficients ofthe polynomials g and f are the same, with exception of the coefficient ofthe monomial w θ , so the specializations u i, σ ( f ) and u i, σ ( g ) (Sec. 2.1) differif and only if i = 0 and σ = θ . Hence, as Π θ holds, u , θ does not appearin M , , M , , and M , , and M , ( g ) = M , ( f ), M , ( g ) = M , ( f ), and M , ( g ) = M , ( f ). The specialization of u , θ is a ring homomorphism, so u , θ ( g ) = u , θ ( f ) − f w θ ( α ). By Π θ , u , θ only appears in the diagonal of M , . Hence, M , ( g ) = M , ( f ) − f w θ ( α ) · I , where I is the identity matrix.Therefore, M ( g ) = h M , M , M , M , i ( f ) − f w θ ( α ) · h I i . By construction g ( α ) = 0, α ∈ V P ( f ), thus α ∈ V P ( g ), and so Res P ( g )vanishes. By property Π θ , det( M ) is a multiple of Res P ( d , . . . , d n ), hence M ( g ) is singular. Let v ∈ ker( M ( g )), then M ( g ) · v = 0 ⇐⇒ h M , M , M , M , i ( f ) · v = f w θ ( α ) · [ I ] · v . Multiplying this equality by the non-singular matrix related to the Schurcomplement of M , ( f ), h I − M , · M − , I i ( f ), we obtain h M , M , M , − M , · M − , · M , ) i ( f ) · v = f w θ ( α ) · [ I ] · v . Consider the lower part of the matrices in the previous identity, h M , − M , · M − , · M , i ( f ) · v = f w θ ( α ) · h I i · v and let ¯ v := h I i · v be a truncation of the vector v . Then,( M , − M , · M − , · M , )( f ) · ¯ v = f w θ ( α ) · ¯ v . This equality proves that f w θ ( α ) is an eigenvalue of the Schur complementof M , ( f ) with eigenvector ¯ v . 16et f ∈ S ( d ) × · · · × S ( d n ), Eq. (1), be a square system. Consider f ∈ S ( d ) and θ ∈ A ( d ). We say that the rational function f w θ separatesthe zeros of the system, if for all α ∈ V P ( f ), w θ ( α ) = 0 and for all α, α ′ ∈ V P ( f , . . . , f n ), f w θ ( α ) = f w θ ( α ′ ) ⇐⇒ α = α ′ . Corollary 4.3.
Under the assumptions of Thm. 4.2, if the row dimensionof M , is MHB ( d , . . . , d n ) , f w θ separates the zeros of ( f , . . . , f n ) and thereare MHB ( d , . . . , d n ) different solutions for this subsystem (over P ), then theSchur complement of M , ( f ) is diagonizable with eigenvalues f w θ ( α ) , for α ∈ V P ( f , . . . , f n ) .Proof. As a consequence of Thm. 4.2, for each α ∈ V P ( f ) we have an eigen-value f w θ ( α ) for the Schur complement of M , ( f ). As f w θ separates these ze-ros, all the eigenvalues are different. Hence, we have as many different eigen-values as the dimension of the matrix, so the matrix is diagonalizable.Note that, as the MHB bounds the number of isolated solutions countingmultiplicities, we can not use Thm. 4.3 when we have a square system f suchthat its solutions over P have multiplicities. Lemma 4.4.
Under the assumptions of Thm. 4.2, assume that
Res P ( f ) = 0 and det( M ) = q · Res P ( d , . . . , d n ) , where q is a non-zeroconstant in K . If λ is an eigenvalue of the Schur complement of M , ( f ) ,then there is α ∈ V P ( f ) such that λ = f w θ ( α ) .Proof. Consider the system g := (( f − λ · w θ ) , f , . . . , f n ). As the matrixof the Schur complement in the proof of 4.2 is invertible, we extend ¯ v to v = (cid:2) M − , · M , I (cid:3) ( f ) ¯ v , and reverse the argument in this proof to show that M ( g ) is singular. As the determinant of M is a non-zero constant multipleof the resultant, we deduce that Res P ( g ) is zero. Let α ∈ V P ( g ), then α ⊂ V P ( f ) and ( f − λ · w θ )( α ) = 0, equivalently, f ( α ) = λ · w θ ( α ). As weassumed that Res P ( f ) = 0, then f ( α ) = 0 and so f w θ ( α ) = λ . Proposition 4.5.
Under the assumptions of Thm. 4.2, assume det( M ) = q · Res P ( d , . . . , d n ) , where q is a non-zero constant in K , and that the (row) di-mension of M , is MHB ( d , . . . , d n ) . Then for any system f := ( f , . . . , f n ) , V P ( w θ , f , . . . , f n ) = ∅ if and only if M , ( f ) is non-singular.Proof. Consider the determinant of M . As it is a multiple of the resul-tant (Sec. 2.1) and the resultant is a multihomogeneous polynomial of de-gree MHB ( d , . . . , d n ) with respect to u , we can write det( M ) = P ( u ) · MHB ( d ,..., d n )0 , θ + Q ( u ), where P ( u ) ∈ K [ u ] does not involve the variables in u and Q ( u ) ∈ K [ u ] is a polynomial such that none of its monomials aremultiple of u MHB ( d ,..., d n )0 , θ . As Π θ holds, u , θ only appears in the diagonal of M , . Consider the expansion by minors of det( M ). If the (row) dimen-sion of M , is MHB ( d , . . . , d n ), then P ( u ) = ± det( M , ). The polynomial P ( u ) is a constant multiple of the cofactor of u MHB ( d ,..., d n )0 , θ in the resultantRes P ( d , . . . , d n ).By construction, Q ( u ) is a homogeneous polynomial with respect to thevariables u of degree MHB ( d , . . . , d n ). As u MHB ( d ,..., d n )0 , θ does not divide anymonomial in Q ( u ), each monomial involves a variables of u different to u ,θ .Hence, for any system f , we have Q ( w θ , f , . . . , f n ) = 0. By construction,the polynomial P ( u ) does not involve any of the variables of u . Thereforedet( M , )( f ) = det( M , )( w θ , f , . . . , f n ). Therefore, for any system f , q · Res P ( d , . . . , d n )( w θ , f , . . . , f n ) = det( M )( w θ , f . . . f n ) = ± det( M , )( w θ , f . . . f n ) = ± det( M , )( f ). The determinant of M is anon-zero constant multiple of the resultant, hencedet( M , )( f ) = 0 if and only if the system ( w θ , f , . . . , f n ) has no solutionsover P , i.e., V P ( w θ , f , . . . , f n ) = ∅ .If the square system f = ( f , . . . , f n ) has no solutions at infinity in P ,that is all the coordinates of the solutions are not zero, then the evaluationof the solutions of f at any monomial in S ( d ) is not zero. Hence, for any w θ ∈ S ( d ), V P ( w θ , f , . . . , f n ) = ∅ . By Prop. 4.5, M , ( f , f , . . . , f n ) isinvertible. To avoid solutions at infinity, in the 0-dimensional multihomoge-neous case, we perform a generic linear change of coordinates that preservesthe multihomogeneous structure. We state the following corollary withoutproof. Corollary 4.6.
Consider a square multihomogeneous system f ∈ S ( d ) ×· · · × S ( d n ) with finite V P ( f ) . Choose θ ∈ A ( d ) and let M be a resultantmatrix for Res P ( d , . . . , d n ) , such that Π θ holds. Consider any f ∈ S ( d ) .Then, for a generic linear change of coordinates A , preserving the multiho-mogeneous structure, the matrix M , ( f , f ◦ A, . . . , f n ◦ A ) is invertible. We can use Thm. 4.2 to solve the 2-bilinear systems.
Theorem 4.7.
Assume a 2-bilinear system f , . . . , f n of type ( n x , n y , n z ; r, t ) , such that V P ( f , . . . , f n ) is finite. Choose θ ∈ A ( d ) and con-sider the M be the matrix of δ ( F (2) , m ) (Sec. 3.2) for the “universal” system (2) rearranged with respect to the monomial w θ . Choose f ∈ S (1 , , .Then, after applying a generic linear change of coordinates A , preservingthe multihomogeneous structure, the eigenvalues of the Schur complement of M , ( f , f ◦ A, . . . , f n ◦ A ) are the evaluations of f w θ over V P ( f ◦ A, . . . , f n ◦ A ) .Proof. We only need to check if the Koszul resultant matrix has the propertyΠ θ . The entries of our matrix are the variables of u up to sign. Note that if u i,σ ∈ u appears in an entry, then it does not appear in the other entries in thesame row, or column. Hence, we can rearrange the matrix in such a way thatthe coefficient u ,θ only appears in the diagonal of M , . As the determinantof the system is a constant multiple of the resultant, the dimension of M , the degree of u in the determinant, which equals the MHB . Example 4.8 (Cont.) . In the previous example (Ex. 3.5), we choose θ =((1 , , (1 , , (1 , ∈ A (1 , , and partition the matrix as (cid:2) M , M , M , M , (cid:3) . If weconsider the Schur complement, we get (cid:2) − − (cid:3) . The characteristic polynomialof this matrix is X − X + 3 , whose roots are f w θ ( α ) = 3 and f w θ ( α ) = 1 . We fix θ ∈ A ( d ). We consider the degree vector m = ( n y − , − , n x + n y − r +1) and the determinantal formula M for the map δ ( F (2) , m ) (Sec. 3.2). Westudy the right eigenvectors of the Schur complement of M , to recover thecoordinates of all the solutions of a 2-bilinear system f of type ( n x , n y , n z ; r, s )(Sec. 2.2). We assume that the number of different solutions is V P ( f ) = MHB ( f ).We augment f to f by adding a trilinear polynomial f , which we specifyin the sequel. We study the right eigenvalues of the Schur complement of M , ( f ). We reduce the analysis of the kernel of δ ( f , m ) to the analysis ofa map in a strand of the Koszul complex of a system with common solutions.Let α = ( α x , α y , α z ) ∈ P , and without loss of generality assume that α t, = 0, for t ∈ { x, y, z } . First, we study the kernel of δ ( f , m ), whenthe overdetermined system f has common solutions. We relate this kernelto the eigenvectors, as we did in the proof of thm. 4.2. For each variable t ∈ { x, y, z } , consider the dual form tα ( d t ) := X θ t ∈A ( d t ) t θ t t d t ( α t ) ∂t θ ∈ S t ( d t ) ∗ d t ≥
0. If d t <
0, then we take tα ( d t ) := 0. Observation 4.9.
For each variable t ∈ { x, y, z } , given a polynomial g t ∈ S t ( ¯ d t ) , such that ¯ d t ≤ d t , then operator ⋆ t , Eq. (9) , acts over g t and tα ( d t ) as the evaluation of g t t ¯ dt at α , that is g t ⋆ t tα ( d t ) = g t t ¯ d t ( α t ) · tα ( d t − ¯ d t ) . To simplify notation, given f ∈ S ( d x , d y , d z ) and ( α x , α y , α z ) ∈ P , wedenote by f ( α x , α y ) ∈ S z ( d z ) the partial evaluation of fx dx y dy at x = α x and y = α y . This evaluation is well-defined because the numerator anddenominator share the same degrees w.r.t. x and y . Lemma 4.10.
Consider d = ( d x , d y , d z ) , ¯ d = ( ¯ d x , ¯ d y , ¯ d z ) . Let f ∈ S ( ¯ d ) and g z ∈ S z ( d z ) . If d x ≥ ¯ d x and d y ≥ ¯ d y , then the map ψ (Eq. (10) ) acts over xα ( d x ) ⊗ yα ( d y ) ⊗ g z and f , as the multiplication of g z and f ( α x , α y ) , that is ψ ( xα ( d x ) ⊗ yα ( d y ) ⊗ g z , f ) = xα ( d x − ¯ d x ) ⊗ yα ( d y − ¯ d y ) ⊗ (cid:0) g z · f ( α x , α y ) (cid:1) . Let ω (1) := { I : e I ∈ V r,s − n z +1 , E } and ω (2) := { J : e J ∈ V r,s − n z , E } . Let ρ α : K ω (1) × K ω (2) → L , ⊕ L , , Eq. (7), ρ α ( λ (1) , λ (2) ) := X I ∈ ω (1) λ (1) I · (cid:16) xα (1) ⊗ yα ( r − n y ) ⊗ ⊗ e I (cid:17) + X J ∈ ω (2) λ (2) J · (cid:16) xα (1) ⊗ yα ( r − n y + 1) ⊗ ⊗ e J (cid:17) As ω (1) + ω (2) = (cid:0) s +1 s − n z +1 (cid:1) , we write ρ α : K ( s +1 s − nz +1 ) → K . Lemma 4.11.
The linear map δ ( f , m ) ◦ ρ α : K ( s +1 s − nz +1 ) → K is equivalentto the ( s − n z + 1) -th map of the Koszul complex of the following system,consisting of s + 1 linear polynomials in z , f z := (cid:16) f ( α x , α y ) , f r +1 ( α x , α y ) , . . . , f n ( α x , α y ) (cid:17) , (11) restricted to its 0-graded part, i.e. the strand of the Koszul complex such thatits ( s − n z + 1) -th module is isomorphic to K ( s +1 s − nz +1 ) . f has a solution ( α x , α y , α z ) ∈ V P ( f ), then, α z is a solution of the linearsystem f z , that is α z ∈ V P ( f z ). As f z is an overdetermined system, theKoszul complex f z is not exact [Lan02, Thm. XXI.4.6]. Lemma 4.12.
Let f be an overdetermined 2-bilinear system. If α ∈ V P ( f ) ,then there is a non-zero b λ α ∈ K ( s +1 s − nz +1 ) such that δ ( f , m ) ◦ ρ α ( b λ α ) = 0 .Proof. Following Lem. 4.11, if we compose δ ( f , m ) and ρ α , then we obtaina map which is similar to the 0-graded part of the ( s − n z + 1)-th map of theKoszul complex of the s + 1 linear polynomials in z , f z , Eq. (11). As thelinear system f z has a solution α z , at most n z of its polynomials are linearlyindependent. Hence, the Koszul complex of f z is isomorphic to a Koszulcomplex K ( e f , . . . , e f n z , , . . . ,
0) of a system of s +1 linear polynomials, where( s +1 − n z ) of them are equal to zero [Lan02, Lem. XXI.4.2]. The ( s +1 − n z )-thmap of K ( e f , . . . , e f n z , , . . . ,
0) maps e n z +1 ∧ . . . ∧ e s +1 − n z to zero. Hence, its 0-graded part has a non-trivial kernel, and so there is a non-zero b λ α ∈ K ( ss − nz +1 )such that δ ( f , m ) ◦ ρ α ( b λ α ) = 0. Theorem 4.13.
Let f = ( f , . . . , f n ) be a square 2-bilinear system of type ( n x , n y , n z ; r, s ) , such that it has (cid:0) rn y (cid:1) · (cid:0) sn z (cid:1) different solutions over P . Con-sider θ ∈ A (1 , , such that Res (2) P ( w θ , f , . . . , f n ) = 0 and f ∈ S (1 , , such that f w θ separates the elements in V P ( f ) . Let m := ( n y − , − , n x + n y − r + 1) and M ∈ K [ u ] K×K related to δ ( F (2) , m ) for the overdetermined 2-bilinear “universal” system (Thm. 3.3). Then, theSchur complement of M , ( f ) is diagonalizable, each eigenvalue is f w θ ( α ) ,for α ∈ V P ( f , . . . , f n ) , and we can extend the eigenvector ¯ v α related to α to v α := (cid:2) M − , · M , I (cid:3) ( f ) · ¯ v α such that v α is the element ρ α ( b λ α ) , for some b λ α ∈ K ( s +1 s − nz +1 ) .Proof. By Cor. 4.3, the Schur complex of M , ( f ) is diagonalizable and everyeigenvalues is different. For each α ∈ V P ( f ), consider the eigenvalue f w θ ( α ),related eigenvector ¯ v α , and the system g α := ( f − f w θ ( α ) , f , . . . , f n ). ByLem. 4.12, there is a λ α ∈ K such that δ ( g α , m ) ◦ ρ ( λ α ) = 0. Hence,there is a w α , representing ρ ( λ α ) = 0, in the kernel of M ( g α ). Following theproof of Thm. 4.2, each element in the kernel of the Schur complement of21 , ( g α ) is related to an eigenvector of the Schur complement of M , ( f )with corresponding eigenvalue f w θ ( α ). As for each eigenvalue we have onlyone eigenvector, then the dimension of this kernel is 1. Hence, the truncationof w α , ¯ w α := (0 | I ) · w α , is a multiple of ¯ v α , where 0 is the zero matrix ofappropriate dimension.As M , ( g α ) is invertible and M ( g α ) · w α = 0, it holds that (cid:2) M − , · M , I (cid:3) ( g α ) ¯ w α = w α . As (cid:2) M − , · M , I (cid:3) ( g α ) does not involve u ,θ , then (cid:2) M − , · M , I (cid:3) ( g α ) = (cid:2) M − , · M , I (cid:3) ( f ). Therefore, we conclude that, as ¯ v α is amultiple of ¯ w α , then v α = (cid:2) M − , · M , I (cid:3) ( f ) · ¯ v α is a multiple of w α .In the following example we use Thm. 4.13 to recover α . Example 4.14 (Cont.) . The eigenvalue of f w θ ( α ) = 1 is ¯ v α := (1 , ⊤ . Byextending ¯ v α , we get v α := (cid:2) M − , · M , I (cid:3) ( f ) · (cid:0) (cid:1) = (4 , , , , , , , , , ⊤ which represents ρ α (1 ,
1) = (cid:16) ∂x (1 , + 3 ∂x (0 , (cid:17) ⊗ (cid:16) ∂y (2 , + 2 ∂y (1 , + 4 ∂y (0 , (cid:17) ⊗ ⊗ e { , , } + (cid:16) ∂x (1 , + 3 ∂x (0 , (cid:17) ⊗ (cid:16) ∂y (1 , + 2 ∂y (0 , (cid:17) ⊗ ⊗ e { , , } Hence, xα (1) = (cid:16) ∂x (1 , + 3 ∂x (0 , (cid:17) , and so α ,x = (1 : 3) ∈ P . Also, yα (1) = (cid:16) ∂y (1 , + 2 ∂y (0 , (cid:17) , and then α ,y = (1 : 2) ∈ P . We note that yα (2) = (cid:16) · · ∂y (2 , + 1 · · ∂y (1 , + 2 · · ∂y (0 , (cid:17) .We can recover α ,z as the solution of f ( α ,x , α ,y , z ) = 0 , f ( α ,x , α ,y , z ) = 0 f ( α ,x , α ,y , z ) = 0 f ( α ,x , α ,y , z ) = − z + 3 z Hence, α ,z = (1 : 3) ∈ P and so α = (1 : 3 ; 1 : 2 ; 1 : 3) ∈ P . Size of matrices and FGb
As there are no tight bounds for the complexity of Gr¨obner basis algorithmsfor solving 2-bilinear systems, we compare against our algorithms experi-mentally in Table 1. We consider the state-of-the-art Gr¨obner basis imple-mentation, FGb [Fau10]. For each set of parameters, we consider a randomsquare 2-bilinear system and we dehomogenize the system to compute itsGr¨obner basis. We compared the ratio between the size of the maximal ma-trix appearing in the Gr¨obner basis computation and the size of our Koszulresultant matrix, for all the cases n ≤
15. For reasons of space we onlypresent some indicative examples for n = 12. The rest of the cases can befound in . The resultsare promising and motivate the study of the structure Koszul resultant ma-trix to develop algorithms for faster linear algebra with such matrices.Table 1: Matrix sizes and ratios of Koszul matrix and FGb. n x n y n z r s Size δ Size FGb Ratio2 6 4 7 5 630 ×
630 1769 × . ∼
10 1 1 10 2 352 ×
352 709 ×
422 2 . ∼ × × . ∼ × × . ∼ × × / . ∼ × × / . ∼ × × / ∼ Acknowledgments:
We thank Laurent Bus´e and Carlos D’Andrea forhelpful discussions and references, and the anonymous reviewers for the com-ments and suggestions. The authors are partially supported by ANR JCJCGALOP (ANR-17-CE40-0009) and the PGMO grant GAMMA.23 eferences [AS88] Winfried Auzinger and Hans J Stetter. An elimination algo-rithm for the computation of all zeros of a system of multivari-ate polynomial equations. In
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Proof of Lem. 4.10.
Consider f = P σ c σ x σ x y σ y z σ z . As ψ is a bilinear mapand the tensor product is multilinear, it is enough to prove this lemma onlyfor the monomials x σ x y σ y z σ z ∈ S ( ¯ d )., ψ ( xα ( d x ) ⊗ yα ( d y ) ⊗ g, f ) = P σ c σ ψ ( xα ( d x ) ⊗ yα ( d y ) ⊗ g, x σ x y σ y z σ z ). For that reason, we study the monomial case, ψ ( xα ( d x ) ⊗ yα ( d y ) ⊗ g z , x σ x ⊗ y σ y ⊗ z σ z ) = (cid:16) x σ x ⋆ x xα ( d x ) (cid:17) ⊗ (cid:16) y σ y ⋆ y yα ( d y ) (cid:17) ⊗ (cid:16) g z · z σ z (cid:17) = (cid:16) x σ x x ¯ d x ( α x ) xα ( d x − ¯ d x ) (cid:17) ⊗ (cid:16) y σ y y ¯ d y ( α y ) yα ( d y − ¯ d y ) (cid:17) ⊗ (cid:16) g z · z σ z (cid:17) = (cid:16) xα ( d x − ¯ d x ) (cid:17) ⊗ (cid:16) yα ( d y − ¯ d y ) (cid:17) ⊗ (cid:16) g z · x σ x x ¯ d x ( α x ) y σ y y ¯ d y ( α y ) · z σ z (cid:17) Then, we have ψ ( xα ( d x ) ⊗ yα ( d y ) ⊗ g, f ) = X σ c σ ( xα ( d x − ¯ d x ) ⊗ yα ( d y − ¯ d y ) ⊗ g z · x σ x x ¯ d x ( α x ) · y σ y y ¯ d y ( α y ) · z σ z ) xα ( d x − ¯ d x ) ⊗ yα ( d y − ¯ d y ) ⊗ g z · X σ c σ x σ x x ¯ d x ( α x ) · y σ y y ¯ d y ( α y ) · z σ z xα ( d x − ¯ d x ) ⊗ yα ( d y − ¯ d y ) ⊗ g z · f ( α x , α y ) Proof of Lem. 4.11.
We split the map ρ as ρ ( λ (1) , λ (2) ) := ρ (1) α ( λ (1) )+ ρ (2) α ( λ (2) ),where ρ (1) α : K ω (1) → L , , Eq. (7), such that, ρ (1) α ( λ (1) ) := X I ∈ ω (1) (cid:16) xα (1) ⊗ yα ( r − n y ) ⊗ λ (1) I ⊗ e I (cid:17) , and ρ (2) α : K ω (2) → L , , Eq. (7), such that ρ (2) α ( λ (2) ) := X J ∈ ω (2) (cid:16) xα (1) ⊗ yα ( r − n y + 1) ⊗ λ (2) J ⊗ e J (cid:17) . δ ( f , m ) ◦ ρ α = δ ( f , m ) ◦ ρ (1) α + δ ( f , m ) ◦ ρ (2) α , we study δ ( f , m ) ◦ ρ (1) α and δ ( f , m ) ◦ ρ (2) α separately.Following the definition of δ (Sec. 3.2) we have δ ( f , m ) ◦ ρ (1) α = X I ∈ ω (1) λ (1) I δ ( f , m ) (cid:16) xα (1) ⊗ yα ( r − n y ) ⊗ ⊗ e I (cid:17) = X I ∈ ω (1) λ (1) I (cid:16) r X i =1 ( − i − ψ ( xα (1) ⊗ yα ( r − n y ) ⊗ , f I i ) ⊗ e I \{ I i } + n x + n y +1 X i = r +1 ( − i − ψ ( xα (1) ⊗ yα ( r − n y ) ⊗ , f I i ) ⊗ e I \{ I i } (cid:17) . By Lem. 4.10 we have, δ ( f , m ) ◦ ρ (1) α = X I ∈ ω (1) λ (1) I (cid:16) r X i =1 ( − i − xα (0) ⊗ yα ( r − n y − ⊗ f I i ( α x , α y ) ⊗ e I \{ I i } + n x + n y +1 X i = r +1 ( − i − xα (0) ⊗ yα ( r − n y ) ⊗ f I i ( α x , α y ) ⊗ e I \{ I i } (cid:17) . For i ≤ r , f I i ∈ S (1 , , f I i ( α x , α y ) = f I i ( α ) = 0. δ ( f , m ) ◦ ρ (1) α = X I ∈ ω (1) λ (1) I n x + n y +1 X i = r +1 ( − i − xα (0) ⊗ yα ( r − n y ) ⊗ f I i ( α x , α y ) ⊗ e I \{ I i } = xα (0) ⊗ yα ( r − n y ) ⊗ (cid:0) X I ∈ ω (1) n x + n y +1 X i = r +1 ( − i − λ (1) I f I i ( α x , α y ) ⊗ e I \{ I i } (cid:1) . We conclude that the image of δ ( f , m ) ◦ ρ (1) α belongs to L , .Now consider δ ( f , m ) ◦ ρ (2) α . Following a similar procedure, we deduce29 ( f , m ) ◦ ρ (2) α = xα (0) ⊗ yα ( r − n y ) ⊗ X I ∈ ω (2) (cid:16) λ (2) I f ( α x , α y ) ⊗ e I −{ } (cid:17) + xα (0) ⊗ yα ( r − n y − ⊗ X I ∈ ω (2) r +1 X i =2 (cid:16) ( − i − λ (2) I f I i ( α x , α y ) ⊗ e I −{ I i } (cid:17) + xα (0) ⊗ yα ( r − n y + 1) ⊗ X I ∈ ω (2) n x + n y +1 X i = r +1 (cid:16) ( − i − λ (2) I f I i ( α x , α y ) ⊗ e I −{ I i } (cid:17) For 1 ≤ i ≤ r + 1, f I i ∈ S (1 , , f I i ( α x , α y ) = f I i ( α ) = 0. Hence, δ ( f , m ) ◦ ρ (2) α = xα (0) ⊗ yα ( r − n y ) ⊗ X I ∈ ω (2) (cid:16) λ (2) I f ( α x , α y ) ⊗ e I −{ } (cid:17) + xα (0) ⊗ yα ( r − n y + 1) ⊗ X I ∈ ω (2) n x + n y +1 X i = r +1 (cid:16) ( − i − λ (2) I f I i ( α x , α y ) ⊗ e I −{ I i } (cid:17) Therefore, the image of δ ( f , m ) ◦ ρ (2) α belongs to L , ⊕ L , .We can rewrite δ ( f , m ) ◦ ρ α : K ( s +1 s − nz +1 ) → L , ⊕ L , as( δ ( f , m ) ◦ ρ α )( λ ) = xα (0) ⊗ yα ( r − n y ) ⊗ P ( λ )+ xα (0) ⊗ yα ( r − n y + 1) ⊗ ( − r P ( λ )where P ( λ ) := X I ⊂ ω (2) λ I f ( α x , α y ) ⊗ e I \{ } + X J ⊂ ω (1) s − n z +1 X j =1 ( − j − λ J f J j ( α x , α y ) ⊗ e J \{ J j } P ( λ ) := X I ⊂ ω (2) s − n z X j =2 ( − r + j − λ I f I j ( α x , α y ) ⊗ e I \{ I j }
30e observe that the intersection between the image of P and − P is triv-ial, because Im( P ) ∈ S z (1) ⊗ V r,s − n z , E and Im( P ) ∈ S z (1) ⊗ V r,s − n z − , E .Hence, P + P vanishes if and only if P and P vanish. Hence, δ ◦ ρ α isequivalent to the map λ P ( λ ) + P ( λ ). Note that, for all I ∈ ω (1) ∪ ω (2) , { , . . . , r } ⊂ I . Therefore, if we expand this map we conclude that it isequivalent to the 0-graded part of the ( s − n z + 1)-th map of the Koszulcomplex of the linear system f z . P ( λ ) + P ( λ ) = X J ⊂{ ,r +1 ,...,n } J = s − n z +1 s − n z +1 X j =1 ( − j − λ J f J j ( α x , α y ) ⊗ e { ...r }∪ J \{ J j }}