Showing papers on "Generic polynomial published in 2020"

Whitney theorem for complex polynomial mappings

Abstract: For given natural numbers $$d_1,d_2$$ we describe the topology of a generic polynomial mapping $$F=(f,g):X\rightarrow {{\mathbb {C}}}^2$$ , with deg $$f\le d_1$$ and deg $$g\le d_2$$ . Here X is a complex plane or a complex sphere.

Optimal density for values of generic polynomial maps

Abstract: We establish that the optimal bound for the size of the smallest integral solution of the Oppenheim Diophantine approximation problem $|Q(x)-\\xi|<\\epsilon$ for a generic ternary form $Q$ is $|x|\\ll\\epsilon^\{-1\}$. We also establish an optimal rate of density for the values of polynomials maps in a number of other natural problems, including the values of linear forms restricted to suitable quadratic surfaces, and the values of the polynomial map defined by the generators of the ring of conjugation-invariant polynomials on $M_3(\\Bbb\{C\})$. These results are instances of a general approach that we develop, which considers a rational affine algebraic subvariety of Euclidean space, invariant and homogeneous under an action of a semisimple Lie group $G$. Given a polynomial map $F$ defined on the Euclidean space which is invariant under a semisimple subgroup $H$ of the acting group $G$, consider the family of its translates $F\\circ g$ by elements of the group. We study the restriction of these polynomial functions to the integer points on the variety confined to a large Euclidean ball. Our main results establish an explicit rate of density for their values, for generic polynomials in the family. This problem has been extensively studied before when the polynomials in question are linear, in the context of classical Diophantine approximation, but very little was known about it for polynomial of higher degree. We formulate a heuristic pigeonhole lower bound for the density and an explicit upper bound for it, formulate a sufficient condition for the coincidence of the lower and upper bounds, and in a number of natural examples establish that they indeed match. Finally, we also establish a rate of density for values of homogeneous polynomials on homogeneous projective varieties.

On Polynomial Modular Number Systems over $\mathbb{Z}/p\mathbb{Z}$

Abstract: Since their introduction in 2004, Polynomial Modular Number Systems (PMNS) have become a very interesting tool for implementing cryptosystems relying on modular arithmetic in a secure and efficient way. However, while their implementation is simple, their parameterization is not trivial and relies on a suitable choice of the polynomial on which the PMNS operates. The initial proposals were based on particular binomials and trinomials. But these polynomials do not always provide systems with interesting characteristics such as small digits, fast reduction, etc. In this work, we study a larger family of polynomials that can be exploited to design a safe and efficient PMNS. To do so, we first state a complete existence theorem for PMNS which provides bounds on the size of the digits for a generic polynomial, significantly improving previous bounds. Then, we present classes of suitable polynomials which provide numerous PMNS for safe and efficient arithmetic.

Generic Newton polygon for exponential sums in n variables with parallelotope base

Abstract: Let $p$ be a prime number. Every $n$-variable polynomial $f(\underline x)$ over a finite field of characteristic $p$ defines an Artin--Schreier--Witt tower of varieties whose Galois group is isomorphic to $\mathbb{Z}_p$. Our goal of this paper is to study the Newton polygon of the $L$-function associated to a finite character of $\mathbb{Z}_p$ and a generic polynomial whose convex hull is an $n$-dimensional paralleltope $\Delta$. We denote this polygon by $\mathrm{GNP}(\Delta)$. We prove a lower bound of $\mathrm{GNP}(\Delta)$, which is called the improved Hodge polygon $\mathrm{IHP}(\Delta)$. We show that $\mathrm{IHP}(\Delta)$ lies above the usual Hodge polygon $\mathrm{HP}(\Delta)$ at certain infinitely many points, and when $p$ is larger than a fixed number determined by $\Delta$, it coincides with $\mathrm{GNP}(\Delta)$ at these points. As a corollary, we roughly determine the distribution of the slopes of $\mathrm{GNP}(\Delta)$.

Polynomial composites and certain types of fields extensions

Abstract: In this paper I consider polynomial composites with the coefficients from $K\subset L$. We already know many properties, but we do not know the answer to the question of whether there is a relationship between composites and field extensions. I present the characterization of some known field extensions in terms of polynomial composites. This paper contains the opening problem of characterization of ideals in polynomial composites with respect to various field extensions. I also present the full possible characterization of certain field extensions.

Universal Decomposition Algebras Represent Endomorphisms

Abstract: The goal of this paper is to supply an explicit description of the universal decomposition algebra of the generic polynomial of degree $n$ into the product of two monic polynomials, one of degree $r$, as a representation of Lie algebras of $n\times n$ matrices with polynomial entries. This is related with the bosonic vertex representation of the Lie algebra $gl_\infty$ due to Date, Jimbo, Kashiwara and Miwa.

On generic polynomial differential equations of second order on the circle

Abstract: In the space of second-order differential equations with right-hand sides that are polynomials of degree n  , an open everywhere dense set is distinguished, consisting of equations that are structurally stable on the Poincaré circle and on the projective plane.

Characterizing Triviality of the Exponent Lattice of A Polynomial through Galois and Galois-Like Groups

Abstract: The problem of computing \emph{the exponent lattice} which consists of all the multiplicative relations between the roots of a univariate polynomial has drawn much attention in the field of computer algebra. As is known, almost all irreducible polynomials with integer coefficients have only trivial exponent lattices. However, the algorithms in the literature have difficulty in proving such triviality for a generic polynomial. In this paper, the relations between the Galois group (respectively, \emph{the Galois-like groups}) and the triviality of the exponent lattice of a polynomial are investigated. The $\bbbq$\emph{-trivial} pairs, which are at the heart of the relations between the Galois group and the triviality of the exponent lattice of a polynomial, are characterized. An effective algorithm is developed to recognize these pairs. Based on this, a new algorithm is designed to prove the triviality of the exponent lattice of a generic irreducible polynomial, which considerably improves a state-of-the-art algorithm of the same type when the polynomial degree becomes larger. In addition, the concept of the Galois-like groups of a polynomial is introduced. Some properties of the Galois-like groups are proved and, more importantly, a sufficient and necessary condition is given for a polynomial (which is not necessarily irreducible) to have trivial exponent lattice.

A generic algorithm for the identity problem in finite groups and monoids

Abstract: Kapovich, Myasnikov, Schupp and Shpilrain in 2003 developed generic approach to algorithmic problems, which considers an algorithmic problem on "most" of the inputs instead of the entire domain and ignores it on the rest of inputs. This approach can be applied to algorithmic problems, which are hard in the classical sense. The problem of checking identities in algebraic structures is the one of the most fundamental problem in algebra. For finite algebraic structures this problem can be decidable in polynomial time, or hard (co-NP-complete). In this paper we present a generic polynomial algorithm for the identity problem in all finite groups and monoids with elements of period greater than 1. The work is supported by Mathematical Center in Akademgorodok, the agreement with Ministry of Science and High Education of the Russian Federation number 075-15-2019-1613.

Characterizing Triviality of the Exponent Lattice of a Polynomial Through Galois and Galois-Like Groups

Abstract: The problem of computing the exponent lattice which consists of all the multiplicative relations between the roots of a univariate polynomial has drawn much attention in the field of computer algebra. As is known, almost all irreducible polynomials with integer coefficients have only trivial exponent lattices. However, the algorithms in the literature have difficulty in proving such triviality for a generic polynomial. In this paper, the relations between the Galois group (respectively, the Galois-like groups) and the triviality of the exponent lattice of a polynomial are investigated. The \(\mathbb {Q}\)-trivial pairs, which are at the heart of the relations between the Galois group and the triviality of the exponent lattice of a polynomial, are characterized. An effective algorithm is developed to recognize these pairs. Based on this, a new algorithm is designed and implemented to prove the triviality of the exponent lattice of a generic irreducible polynomial, which considerably improves a state-of-the-art implementation of an algorithm of the same type when the polynomial degree becomes larger. In addition, the concept of the Galois-like groups of a polynomial is introduced. Some properties of the Galois-like groups are proved and, more importantly, a sufficient and necessary condition is given for a polynomial (which is not necessarily irreducible) to have trivial exponent lattice.