Multipolynomial resultant algorithms
The paper describes informally algorithms for computing determinants of matrices with polynomial entries from \(Z(x)=Z(x_ 1,\ldots,x_ n)\). In contrast to methods applied in most computer algebra systems the determinants are computed indirectly by a homomorphism/reconstruction method: map the matrix from \(Z(x)\) to \(Z\) by substituting sets of numbers for \((x_ 1,\dots,x_ n)\), compute for each a numerical determinant in \(Z\) and construct the determinant in \(Z(x)\) by interpolation. Several aspects for the interpolation are discussed: A Vandermonde type equation system (using sets \((p_ 1^ k,\dots,p_ n^ k)\) with primes \(p_ i)\), sparse interpolation, probabilistic interpolation and modular evaluation/Chinese remainder reconstruction. The use of the determinant computation for resultants of multivariate polynomial sets is presented in examples: Implication of rational parametric surfaces, real solutions of polynomial equation systems. The theory of multipolynomial resultants and algorithms for their construction are not covered in this article.
algorithms for computing determinants of matrices
scientific article; zbMATH DE number 370646