Related papers: Critical Points and Gr\"obner Bases: the Unmixed C…

This note will study complex polynomial maps of degree $n\ge 2$ with only one critical point.

In this paper we consider systems of partial (multidimensional) linear difference equations. Specifically, such systems arise in scientific computing under discretization of linear partial differential equations and in computational high…

This paper is concerned with linear algebra based methods for solving exactly polynomial systems through so-called Gr\"obner bases, which allow one to compute modulo the polynomial ideal generated by the input equations. This is a topical…

Normalization of polynomials plays a vital role in the approximate basis computation of vanishing ideals. Coefficient normalization, which normalizes a polynomial with its coefficient norm, is the most common method in computer algebra.…

Polynomial system solving is a classical problem in mathematics with a wide range of applications. This makes its complexity a fundamental problem in computer science. Depending on the context, solving has different meanings. In order to…

In this paper, we describe new methods to compute the radical (resp. real radical) of an ideal, assuming it complex (resp. real) variety is finite. The aim is to combine approaches for solving a system of polynomial equations with dual…

Gr\"{o}bner bases are nowadays central tools for solving various problems in commutative algebra and algebraic geometry. A typical use of Gr\"{o}bner bases is the multivariate polynomial system solving, which enables us to construct…

In this article, we study critical points (zeros of derivative) of random polynomials. Take two deterministic sequences $\{a_n\}_{n\geq1}$ and $\{b_n\}_{n\geq1}$ of complex numbers whose limiting empirical measures are same. By choosing…

The purpose of this work is to generalize part of the theory behind Faugere's "F5" algorithm. This is one of the fastest known algorithms to compute a Groebner basis of a polynomial ideal I generated by polynomials f_{1},...,f_{m}. A major…

Answering a question posed by Adam Epstein, we show that the collection of conjugacy classes of polynomials admitting a parabolic fixed point and at most one infinite critical orbit is a set of bounded height in the relevant moduli space.…

This paper studies the concept and the computation of approximately vanishing ideals of a finite set of data points. By data points, we mean that the points contain some uncertainty, which is a key motivation for the approximate treatment.…

This paper presents a conception for computing gr\"{o}bner basis. We convert some of gr\"{o}bner-computing algorithms, e.g., F5, extended F5 and GWV algorithms into a special type of algorithm. The new algorithm's finite termination problem…

Although Buchberger's algorithm, in theory, allows us to compute Gr\"obner bases over any field, in practice, however, the computational efficiency depends on the arithmetic of the ground field. Consider a field $K = \mathbb{Q}(\alpha)$, a…

In this paper, we give decision criteria for normal binomial difference polynomial ideals in the univariate difference polynomial ring F{y} to have finite difference Groebner bases and an algorithm to compute the finite difference Groebner…

In this paper, we apply our minimax theory ([4], [5], [6]) with the one developed by A. Moameni in [2] to formalize a general scheme giving the multiplicity of critical points. Here is a sample of application of the scheme to a critical…

Let $(f\_1,\dots, f\_s) \in \mathbb{Q}\_p [X\_1,\dots, X\_n]^s$ be a sequence of homogeneous polynomials with $p$-adic coefficients. Such system may happen, for example, in arithmetic geometry. Yet, since $\mathbb{Q}\_p$ is not an effective…

Let $\mathbb{K}$ be a field of characteristic zero and $\mathbb{K}[x_1, \dots, x_n]$ the corresponding multivariate polynomial ring. Given a sequence of $s$ polynomials $\mathbf{f} = (f_1, \dots, f_s)$ and a polynomial $\phi$, all in…

Let f be a degree d polynomial defined over the nonarchimedean field C_p, normalized so f is monic and f(0)=0. We say f is post-critically bounded, or PCB, if all of its critical points have bounded orbit under iteration of f. It is known…

Let $\mathbb{R}$ be the field of real numbers. We consider the problem of computing the real isolated points of a real algebraic set in $\mathbb{R}^n$ given as the vanishing set of a polynomial system. This problem plays an important role…

We give a criterion for a collection of polynomials to be a universal Gr\"{o}bner basis for an ideal in terms of the multidegree of the closure of the corresponding affine variety in $(\mathbb{P}^1)^N$. This criterion can be used to give…