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The Parameter Continuation Theorem is the theoretical foundation for polynomial homotopy continuation, which is one of the main tools in computational algebraic geometry. In this note, we give a short proof using Gröbner bases. Our approach gives ...

It is shown that a normalized complex Hadamard matrix of order 6 having three distinct columns each containing at least one $-1$ entry, necessarily belongs to the transposed Fourier family, or to the family of 2-circulant complex Hadamard matrices. ...

The explicit symplectic difference schemes with a number of stages 3 and 5 are considered for the numerical solution of molecular dynamics problems described by systems with separable Hamiltonians. The parameters of the new Forest–Ruth (FR) ...

We are concerned with the arithmetic of solutions to ordinary or partial nonlinear differential equations which are algebraic in the indeterminates and their derivatives. We call these solutions D-algebraic functions, and their equations are ...

In this paper, we examine the structure of systems that are weighted homogeneous for several systems of weights, and how it impacts the computation of Gröbner bases. We present several linear algebra algorithms for computing Gröbner bases for ...

Given polynomials g and f1, …, fp, all in <Formula format="inline"><TexMath><?TeX $\field [x_1,\dots,x_n]$?></TexMath><AltText>Math 1</AltText><File name="issac24-43-inline1" type="svg"/></Formula> for some field <Formula format="inline"><TexMath><?TeX $...

Polyhedral affinoid algebras have been introduced by Einsiedler, Kapranov and Lind in [5] to connect rigid analytic geometry (analytic geometry over non-archimedean fields) and tropical geometry. In this article, we present a theory of Gröbner bases for ...

For a given ideal <Formula format="inline"><TexMath><?TeX $I\subseteq \mathbb {K}[x_1,\dots,x_n,y_1,\dots,y_m]$?></TexMath><AltText>Math 1</AltText><File name="issac24-10-inline1" type="svg"/></Formula> in a polynomial ring with n + m variables, we want ...

Arora & Ge introduced a noise-free polynomial system to compute the secret of a Learning With Errors (LWE) instance via linearization. Albrecht et al. later utilized the Arora-Ge polynomial model to study the complexity of Gröbner basis ...

The explicit symplectic difference schemes are considered for the numerical solution of molecular dynamics problems described by systems with separable Hamiltonians. A general method for finding symplectic schemes of high order of accuracy using ...

• The use of the parametric Gröbner bases and resultants is shown to facilitate the obtaining of new symplectic schemes.
• An efficient two-step optimization algorithm is proposed for obtaining new symplectic schemes.
• 97 new four-stage ...

We introduce the notion of star gluing of numerical semigroups and show that this preserves the arithmetically Cohen-Macaulay and Gorenstein properties of the projective closure. Next, we give a sufficient condition involving Gröbner ...

In this paper, we study the complexity of performing some linear algebra operations such as Gaussian elimination and minimal polynomial computation over an algebraic extension field. For this, we use the theory of Gröbner bases to employ linear ...

The label code of a lattice plays a key role in the characterization of the lattice. Every lattice $\mathrm{\Lambda }$ can be described in terms of a label code L and an orthogonal sublattice ${\mathrm{\Lambda }}^{\prime }$ such that $\mathrm{\Lambda }/{\mathrm{\Lambda }}^{\prime }\cong L$. We identify the binomial ideal associated to an ...${}_{}$${}_{\mathrm{}}{}_{{\mathrm{}}^{}}{}_{}$${}_{\mathrm{}}{}_{{\mathrm{}}^{}}$${}_{}$$\mathrm{}{\mathrm{}}^{}$${}_{}$

A universal analytic Gröbner basis (UAGB) of an ideal of a Tate algebra is a set containing a local Gröbner basis for all suitable convergence radii. In a previous article, the authors proved the existence of finite UAGB’s for polynomial ideals, leaving ...

We describe a recursive algorithm that decomposes an algebraic set into locally closed equidimensional sets, i.e. sets which each have irreducible components of the same dimension. At the core of this algorithm, we combine ideas from the theory of ...

Considering a multivariate polynomial ideal over a given field as a vector space, we investigate for such an ideal a particular linear basis, a so-called staggered linear basis, which contains a Gröbner basis as well. In this paper, we ...

With this paper we present an extension of our recent ISSAC paper about computations of Gröbner(-Shirshov) bases over free associative algebras Z 〈 X 〉. We present all the needed proofs in details, add a part on the direct treatment of ...

The evaluation of a polynomial at several points is called the problem of multi-point evaluation. Sometimes, the set of evaluation points is fixed and several polynomials need to be evaluated at this set of points. Several efficient ...

In this paper we study and relate several invariants connected to the solving degree of a polynomial system. This provides a rigorous framework for estimating the complexity of solving a system of polynomial equations via Gröbner bases ...

In this paper we study the security of the Bluetooth stream cipher E0 from the viewpoint it is a “difference stream cipher”, that is, it is defined by a system of explicit difference equations over the finite field GF ( 2 ). This approach ...