Matrices in elimination theory
This is a survey devoted to recent developments of the theory of resultants and corresponding matrices. The theory of sparse elimination, Newton polytopes, mixed volumes, toric resultant is set forth. Matrices for computing the resultants are considered, generalized Sylvester's, Macaulay's, as well as Newton's matrices and, on the other hand, a generalized Bézout construction (in particular, it is proved that any maximal minor of the Bézout matrix is a multiple of the resultant), combining the methods (Dixon matrices). Comparison of the resultant matrices is made from the computational point of view. Finally, applications to the theory of solving arbitrary systems of polynomial equations are described: monomial bases and multiplication maps, system solving by the \(u\)-resultant, by reduction to the eigenvalue problem [see also \textit{D. Cox, J. Little} and \textit{D. O'Shea}, ``Using algebraic geometry'', Graduate Texts in Math. 185 (1998; Zbl 0920.13026) sometimes for more details], degenerate cases. Some examples are used for illustration. A very wide list of references is enclosed.
scientific article; zbMATH DE number 1369538
scientific article