相关论文: An algorithm to obtain linear determinantal repres…

We give two algorithms to compute linear determinantal representations of smooth plane curves of any degree over any field. As particular examples, we explicitly give representatives of all equivalence classes of linear determinantal…

In this note, we study linear determinantal representations of smooth plane cubics over finite fields. We give an explicit formula of linear determinantal representations corresponding to rational points. Using Schoof's formula, we count…

Can a smooth plane cubic be defined by the determinant of a square matrix with entries in linear forms in three variables? If we can, we say that it admits a linear determinantal representation. In this paper, we investigate linear…

Every real hyperbolic form in three variables can be realized as the determinant of a linear net of Hermitian matrices containing a positive definite matrix. Such representations are an algebraic certificate for the hyperbolicity of the…

We introduce the DeterminantalRepresentations package for Macaulay2, which computes definite symmetric determinantal representations of real polynomials. We focus on quadrics and plane curves of low degree (i.e. cubics and quartics). Our…

The problem of expressing a specific polynomial as the determinant of a square matrix of affine-linear forms arises from algebraic geometry, optimisation, complexity theory, and scientific computing. Motivated by recent developments in this…

The discriminant of a smooth plane cubic curve over the complex numbers can be written as a product of theta functions. This provides an important connection between algebraic and analytic objects. In this paper, we perform a new approach…

In this paper, we consider the problem of representing a multivariate polynomial as the determinant of a definite (monic) symmetric/Hermitian linear matrix polynomial (LMP). Such a polynomial is known as determinantal polynomial.…

We consider the problem of writing real polynomials as determinants of symmetric linear matrix polynomials. This problem of algebraic geometry, whose roots go back to the nineteenth century, has recently received new attention from the…

We determine the number of ${\mathbb{F}}_q$-rational points of hyperplane sections of classical determinantal varieties defined by the vanishing of minors of a fixed size of a generic matrix, and identify sections giving the maximum number…

We present here algorithms for efficient computation of linear algebra problems over finite fields.

We contribute a new algebraic method for computing the orthogonal projections of a point onto a rational algebraic surface embedded in the three dimensional projective space. This problem is first turned into the computation of the finite…

We prove that the defining equations of the Fermat curves of prime degree cannot be written as the determinant of symmetric matrices with entries in linear forms in three variables with rational coefficients. In the proof, we use a relation…

The paper develops elementary linear algebra methods to compute the determinants of the tensor symmetrizations of quadratic and hermitian forms over fields of good characteristic. Explicit results are given for the partitions $(n)$,…

We study the arithmetic of bitangents of smooth quartics over global fields. With the aid of computer algebra systems and using Elsenhans--Jahnel's results on the inverse Galois problem for bitangents, we show that, over any global field of…

Due to the rapid development of quantum computing, the compact representation of quantum operations based on decision diagrams has been received more and more attraction. Since variable orders have a significant impact on the size of the…

Let k be an imaginary quadratic number field (with class number 1). We describe a new, essentially linear-time algorithm, to list all isomorphism classes of cubic extensions L/k up to a bound X on the norm of the relative discriminant…

We present algorithms for classification of linear codes over finite fields, based on canonical augmentation and on lattice point enumeration. We apply these algorithms to obtain classification results over fields with 2, 3 and 4 elements.…

We give algorithms for computing with divisors on projective curves over finite fields, and with their Jacobians, using the algorithmic representation of projective curves developed by Khuri-Makdisi. We show that many desirable operations…

A survey on algorithms for computing discrete logarithms in Jacobians of curves over finite fields.