相关论文: Macaulay Style Formulas for Sparse Resultants

We compute two parametric determinants in which rows and columns are indexed by compositions, where in one determinant the entries are products of binomial coefficients, while in the other the entries are products of powers. These results…

Subresultant of two univariate polynomials is a fundamental object in computational algebra and geometry with many applications (for instance, parametric GCD and parametric multiplicity of roots). In this paper, we generalize the theory of…

Algorithms for Gaussian process, marginal likelihood methods or restricted maximum likelihood methods often require derivatives of log determinant terms. These log determinants are usually parametric with variance parameters of the…

We present a probabilistic algorithm to compute the product of two univariate sparse polynomials over a field with a number of bit operations that is quasi-linear in the size of the input and the output. Our algorithm works for any field of…

We prove that the polynomial form of the scattering equations is a Macaulay H-basis. We demonstrate that this H-basis facilitates integrand reduction and global residue computations in a way very similar to using a Gr\"obner basis, but…

A polynomial matrix inequality is a formula asserting that a polynomial matrix is positive semidefinite. Polynomial matrix optimization concerns minimizing the smallest eigenvalue of a symmetric polynomial matrix subject to a tuple of…

We present several new and compact formulas for the modified and integral form of the Macdonald polynomials, building on the compact "multiline queue" formula for Macdonald polynomials due to Corteel, Mandelshtam, and Williams. We also…

A fundamental problem in computational algebraic geometry is the computation of the resultant. A central question is when and how to compute it as the determinant of a matrix. whose elements are the coefficients of the input polynomials…

We consider polynomials of a few linear forms and show how exploit this type of sparsity for optimization on some particular domains like the Euclidean sphere or a polytope. Moreover, a simple procedure allows to detect this form of…

To compute solutions of sparse polynomial systems efficiently we have to exploit the structure of their Newton polytopes. While the application of polyhedral methods naturally excludes solutions with zero components, an irreducible…

The article presents a formula expressing Macaulay constants of a numerical polynomial through its minimizing coefficients. From this, we have that Macaulay constants of Kolchin dimension polynomials do not decrease. For the minimal…

We present a deterministic algorithm for computing all irreducible factors of degree $\le d$ of a given bivariate polynomial $f\in K[x,y]$ over an algebraic number field $K$ and their multiplicities, whose running time is polynomial in the…

The decomposition of the Laughlin wave function in the Slater orthogonal basis appears in the discussion on the second-quantized form of the Laughlin states and is straightforwardly equivalent to the decomposition of the even powers of the…

We present several new and compact formulas for the modified and integral form of the Macdonald polynomials, building on the compact "multiline queue" formula for Macdonald polynomials due to Corteel, Mandelshtam and Williams. We also…

We provide, in a 474 pages study, a comprehensive and self-contained treatment of Resultant Theory for a homogeneous system of polynomials with several variables (as many variables as of polynomials). In a non classical way, we use the…

We introduce a non-commutative resultant, for slice regular polynomials in two quaternionic variables, defined in terms of a suitable Dieudonn\'e determinant.We use this tool to investigate the existence of common zeros of slice regular…

Given a way to evaluate an unknown polynomial with integer coefficients, we present new algorithms to recover its nonzero coefficients and corresponding exponents. As an application, we adapt this interpolation algorithm to the problem of…

Computing the determinant of a matrix with the univariate and multivariate polynomial entries arises frequently in the scientific computing and engineering fields. In this paper, an effective algorithm is presented for computing the…

This article presents a new method to compute matrices from numerical simulations based on the ideas of sparse sampling and compressed sensing. The method is useful for problems where the determination of the entries of a matrix constitutes…

We consider the problem of finding a sparse multiple of a polynomial. Given f in F[x] of degree d over a field F, and a desired sparsity t, our goal is to determine if there exists a multiple h in F[x] of f such that h has at most t…