Matrices in Elimination Theory Ioannis Z Emiris Bernard Mourrain INRIA SAGA BP SophiaAntip olis France FirstLastsophiainriafr httpwwwinriafrsaga Fir stL ast Septemb er Contents Intro duction Elimination theory Notation Classical elimination theory n Resultant over P Resultant over a toric variety Matrix formulations Macaulay matrices Newton matrices Incremental Newton matrices Bzout matrices Dixon matrices Comparison b etween dierent matrices Applications of resultant matrices Monomial bases and multiplication maps System solving by the uresultant System solving by eigenvectors Genericity issues Conclusion Abstract The last decade has witnessed the rebirth of resultant metho ds as a p owerful computational to ol for variable elimination and p olynomial system solving In particular the advent of sparse elimination theory and toric varieties has provided ways to exploit the structure of p olynomials encountered in a numb er of scientic and engineering applications On the other hand the Bezoutian reveals itself as an imp ortant to ol in many areas connected to elimination theory and has its own merits leading to new developments in eective algebraic geometry This survey unies the existing work on resultants with emphasis on constructing matrices that generalize the classic matrices named after Sylvester Bzout and Macaulay The prop erties of the dierent matrix formulations are presented including some complexity issues with an emphasis on variable elimination theory We compare toric resultant matrices to Macaulays matrix and further conjecture the generalization of Macaulays exact rational expression for the resultant p olynomial to the toric case A sparse version of an eective Nullstellensatz is directly obtained A new theorem proves that the maximal minor of a Bzout matrix is a nontrivial multiple of the resultant We discuss applications to constructing monomial bases of quotient rings and multiplication maps as well as to system solving by linear algebra op erations Lastly degeneracy issues a ma jor preo ccupation in practice are examined Throughout the presentation examples are used for illustration and op en questions are stated in order to p oint the way to further research Intro duction Resultants provide an essential to ol in constructive algebra and in equation solving The resultant of an overconstrained p olynomial system characterizes the existence of common ro ots as a condition on the input co ecients If we consider the input co ecients as indep endent indeterminates then the solutions lie in a space of dimension n m where n is the numb er of variables and m is the numb er of co ecients The resultant pro jects the solutions to an mdimensional space and is therefore also known as a projection operator Since it eliminates the input variables the resultant is also known as the eliminant of the given system A numb er of metho ds exists for constructing resultant matrices which are matrices whose determinant is the resultant or more generally a nontrivial multiple of it These matrices represent the most ecient way for computing the resultant p olynomial and for solving systems of p olynomial equations by means of the resultant metho d An example of a matrix that gives precisely the resultant is the determinant of the co ecient matrix of n linear p olynomials or the Sylvester matrix of a p olynomial pair Resultant matrices have b een extensively studied around the turn of the century Their determinants have b een known as inertia forms We give a short historical overview of the impact of resultants in eective algebraic geometry We refer the reader to Mui for a more complete historical description The rst ma jor contribution to resultant theory was probably the work of E Bzout B and Euler By combining two univariate p olynomials P x of degree m Qx of degree n m Bezout observed that he can get m linearly indep endent p olynomials of degree m This yields a m by m matrix called Bezoutian matrix by Sylvester whose determinant is the resultant of P and Q originally called la rsultant de P et Q in French see Las It was worked out latter by Cayley who connected it with the p olynomial Qx P y Qy P x x y mn JJ Sylvester preferred to eliminate directly the monomials x x in the multiples i j x P x x Qx in j m of the initial p olynomials and to ok the determinant of the corresp onding m n m n matrix Although contemp orary and related works Jacobi Richelot Cauchy this metho d remain well known as Sylvesters resultant A generalization of the Bezoutian in several variables was used in the work of A Dixon Dix For p olynomials P P P in two variables of the same degree he to ok some co ecients of their multivariate Bezoutian and added some multiples of the initial p olynomials in order to get a square matrix In section we shall come again to this construction The work of FS Macaulay see Mac Mac vdW generalizes the Sylvester construction to the multivariate case which in turn was extended recently to the case of toric variety Indeed the last two decades have witnessed the ourishing of the theory of sparse elimination Ber GKZ a more complete account is given b elow This theory generalizes several results of classical elimination theory on multivariate p olynomial systems by considering the structure of the given p olynomials namely the co ecients which are a priori zero and the supp ort and Newton p olytop e dened by the nonzero co ecients This leads to stronger algebraic and combinatorial results in general whose complexity dep ends on eective rather than total degree The toric or sparse resultant generalizes the classical resultant of n homogeneous p olynomials in n variables in the sense that they coincide when all p olynomial co ecients are nonzero The toric resultant coincides with the Sylvester resultant if the system is comprised of two univariate p olynomials Unlike its classical counterpart however the toric resultant dep ends on the nonzero monomials only and therefore it has lower degree for sparse inputs The renewed interest in elimination theory and the asso ciated matrix metho ds for system solving is man ifold Bezoutians app ear to b e a fundamental to ol in several domains such as residue theory SS Kun and complex analysis BGVY BY AK Dierent kinds of resultant matrices are used in com plexity theory in eective algebraic geometry SS FGS LM and of course elimination theory Jou The use of Bezoutians in imp ortant applications from an algorithmic p oint of view is illus trated in Car BCRS CM For small and mediumsize p olynomial systems corresp onding to zero dimensional varieties the resultant matrix provides one of the most ecient solution metho ds to day This has b een established through a numb er of concrete applications in the forward and inverse kinematics of rob ots and mechanisms as well as the computation of their motion plans Can RR the geometric structure of molecules BMB EMb geometric and solid mo deling graphics and computeraided de sign BGW Hof MD as well as quantier elimination CG Ren Can BPR and the solution of systems of inequalities GV This survey is organized as follows The next section sketches the basic notions of elimination theory starting with notation then the classical theory and nally discusses sparse elimination theory Section is the main part where the dierent matrix formulations are detailed and analyzed Sylvester and Macaulay matrices in section two algorithms for the toric resultant or Newton matrix in sections and Bzout and Dixon matrices in the following two sections and nally a comparison b etween dierent matrix formulations Then we consider how these matrices can b e used for constructing monomial bases multiplica tion maps and ultimately for solving systems of p olynomial equations by dierent metho ds in sections to Section presents metho ds for handling input degeneracy We conclude with a summary Elimination theory Notation n K is the co ecient eld unless otherwise stated it is arbitrary P denotes the pro jective space n K is the of dimension n obtained as a quotient of K nonzero multiple vectors are identied algebraic closure of K K K fg is the eld K except n is the numb er of variables of the p olynomial rings of interest x x denote the n indep endent n variables and x is a shorthand for all of them The notation x represents x x More sets n of variables are sometimes needed and denoted y and u The set of p olynomials in the variables x x with co ecients in K will b e denoted by R n K x x f f are the input p olynomials lying in R K x These can b e Laurent n n p olynomials lying in R K x x Their co ecients are denoted c c ij I J are ideals of R and A R I denotes the quotient algebra of p olynomials p mo dulo the ideal I Z f f stands for the zeroset of f f that is the algebraic variety of p oints K s s n K such that f f If not sp ecied Z f f means the zeroset over the