Matrix formulæ for resultants and discriminants of bivariate tensor-product polynomials

Abstract

The construction of optimal resultant formulae for polynomial systems is one of the main areas of research in computational algebraic geometry. However, most of the constructions are restricted to formulae for unmixed polynomial systems, that is, systems of polynomials which all have the same support. Such a condition is restrictive, since mixed systems of equations arise frequently in many problems. Nevertheless, resultant formulae for mixed polynomial systems is a very challenging problem. We present a square, Koszul-type, matrix, the determinant of which is the resultant of an arbitrary (mixed) bivariate tensor-product polynomial system. The formula generalizes the classical Sylvester matrix of two univariate polynomials, since it expresses a map of degree one, that is, the elements of the corresponding matrix are up to sign the coefficients of the input polynomials. Interestingly, the matrix expresses a primal-dual multiplication map, that is, the tensor product of a univariate multiplication map with a map expressing derivation in a dual space. In addition we prove an impossibility result which states that for tensor-product systems with more than two (affine) variables there are no universal degree-one formulae, unless the system is unmixed. Last but not least, we present applications of the new construction in the efficient computation of discriminants and mixed discriminants.

Koszul-type determinantal formulas for families of mixed multilinear systems

“…However, there are very few results about determinantal formulas for mixed multihomogeneous systems, that is, when the supports are not the same. We know such formulas for scaled multihomogeneous systems [22], that is when the supports are scaled copies of one of them, and for bivariate tensor-product polynomial systems [38,9]. In what follows, we use the Weyman complex to derive new formulas for families of mixed multilinear systems.…”

“…However, when the supports are not the same, that is in the case of mixed multihomogeneous systems, there are very few results. We know determinantal formulas for scaled multihomogeneous systems [22], in which case the supports are scaled copies of one of them, for bivariate tensor-product polynomial systems [9], and for bilinear systems with two different supports [5]. One tool to obtain such formulas is using the Weyman complex [52].…”

Bilinear systems with two supports: Koszul resultant matrices, eigenvalues, and eigenvectors

“…The map δ 1 (f 0 , m) corresponds to a Koszul-type formula, involving multiplication and dual multiplication maps. The matrix that represents this map is a Koszul resultant matrix [MT17,BMT17].…”

“…Another kind of formula is the Koszul-type formula that involves the other maps of the Koszul complex. We call the matrices related to this formula Koszul resultant matri-ces [MT17,BMT17]. For both formulas, the elements of the matrices are linear polynomials in the coefficients of (f 0 , .…”

“…For unmixed multihomogeneous systems, that is systems where all the polynomial 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 scaled multihomogeneous systems [EM12], that is when the supports are scaled copies of one of them, and the bivariate tensor-product case [MT17,BMT17].…”