Matrices in Elimination Theory

Abstract

The last decade has witnessed the rebirth of resultant methods as a powerful computational tool for variable elimination and polynomial system solving. In particular, the advent of sparse elimination theory and toric varieties has provided ways to exploit the structure of polynomials encountered in a number of scientific and engineering applications. On the other hand, the Bezoutian reveals itself as an important tool in many areas connected to elimination theory and has its own merits, leading to new developments in effective algebraic geometry. This survey unifies the existing work on resultants, with emphasis on constructing matrices that generalize the classic matrices named after Sylvester, Bézout and Macaulay. The properties of the different matrix formulations are presented, including some complexity issues, with emphasis on variable elimination theory. We compare toric resultant matrices to Macaulay's matrix and further conjecture the generalization of Macaulay's exact rational expression for the resultant polynomial to the toric case. A new theorem proves that the maximal minor of a Bézout matrix is a non-trivial 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 operations. Lastly, degeneracy issues, a major preoccupation in practice, are examined. Throughout the presentation, examples are used for illustration and open questions are stated in order to point the way to further research.

Computing the common zeros of two bivariate functions via Bézout resultants

“…Monomials are the standard basis in the literature [16,18], due to their simplicity and flexibility for algebraic manipulations. On the other hand, the Chebyshev polynomial basis is a better choice for numerical stability on the real interval [−1, 1] [45].…”

“…There are many different resultant matrices such as Sylvester [16], Bézout [7], and other matrices [5,18,26], and this choice can affect subsequent efficiency (see Table 3.1) and conditioning (see section 5). Usually, resultant matrices are constructed from polynomials expressed in the monomial basis [9,32], but they can be derived when using any other bases, see for example [8].…”

“…These bivariate polynomials were taken from [37], and the degrees are (m p , n p , m q , n q ) = (10,18,8,8). Note that in this example we have taken the domain…”

Circular Cylinders through Four or Five Points in Space

“…We hide a variable (that we consider as a parameter) in order to get a system with one equation more that the number of remaining variables. Then we use a resultant formulation [11], in order to get a condition on this parameter such that the overdetermined system has a solution. From this condition, we deduce the values of the hidden variable for the roots of our system.…”

Resultants and Discriminants for Bivariate Tensor-Product Polynomials

“…ISSAC '17, July [25][26][27][28]2017, Kaiserslautern, Germany and to [27] for various matrix constructions. The optimal construction that one can hope for is a degree-one formula; that is a matrix whose non-zero entries are coefficients of the input polynomials (modulo multiplication with ±1), and whose determinant is equal to the resultant.…”