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We solve the inverse differential Galois problem over differential fields with a large field of constants of infinite transcendence degree over ${\mathbb Q}$. More generally, we show that over such a field, every split differential…
In this paper, we describe a congruence property of solvable polynomials with coefficients in the Gaussian field Q(i).
Let $k$ be a field of characteristic not 2 or 3. We establish polynomial lower bounds on the ambient dimension $N$ for an intersection $X\subset\mathbb{P}^N$ of quadrics, cubics and quartics to have a dense collection of solvable points,…
We define a notion of formal quantum field theory and associate a formal quantum field theory to K-theoretical intersection theories on Hilbert schemes of points on algebraic surfaces. This enables us to find an effective way to compute…
For a finite separable field extension K/k, all subfields can be obtained by intersecting so-called principal subfields of K/k. In this work we present a way to quickly compute these intersections. If the number of subfields is high, then…
This work provides a complete characterization of the solutions of a linear interpolation problem for vector polynomials. The interpolation problem consists in finding n scalar polynomials such that an equation involving a linear…
Some physical applications of the Passare-Tsikh solution of a principal quintic equation are discussed. As an example, a quintic equation of state is solved in detail. This approach provides analytical approximations for several problems…
Deciding whether or not two polynomials have isomoprhic splitting fields over the rationals is the Field Isomorphism Problem. We consider polynomials of the form $f_n(x) = x^4-nx^3-6x^2+nx+1$ with $n \neq 3$ a positive integer and we let…
The purpose of this note is to survey a methodology to solve systems of polynomial equations and inequalities. The techniques we discuss use the algebra of multivariate polynomials with coefficients over a field to create large-scale linear…
We propose an algorithm for determining the irreducible polynomials over finite fields, based on the use of the companion matrix of polynomials and the generalized Jordan normal form of square matrices.
We give a logically and mathematically self-consistent procedure of quantization of free scalar field, including quantization on space-like surfaces. A short discussion of possible generalization to interacting fields is added.
The aim of this paper is to investigate the intersection problem between two linear sets in the projective line over a finite field. In particular, we analyze the intersection between two clubs with eventually different maximum fields of…
In this paper, we derive an explicit combinatorial formula for the number of $k$-subset sums of quadratic residues over finite fields.
We describe a method to evaluate multivariate polynomials over a finite field and discuss its multiplicative complexity.
We propose an approach to the problem of global reconstruction of an orientation field. The method is based on a geometric model called "bisector line fields", which maps a pair of vector fields to an orientation field, effectively…
We generalize Carlitz' result on the number of self reciprocal monic irreducible polynomials over finite fields by showing that similar explicit formula hold for the number of irreducible polynomials obtained by a fixed quadratic…
A method for generating irreducible polynomials of degree n over the finite field GF(2) is proposed. The irreducible polynomials are found by solving a system of equations that brings the information on the internal properties of the…
Let f and g be nonconstant polynomials over an arbitrary field K. In this paper we study the intersection of the polynomial rings K[f] and K[g], and in particular we ask whether this intersection is larger than K. We completely resolve this…
In this paper we will give a unified proof of several results on the sovability of systems of certain equations over finite fields, which were recently obtained by Fourier analytic methods. Roughly speaking, we show that almost all systems…
A general method of obtaining linear differential equations having polynomial solutions is proposed. The method is based on an equivalence of the spectral problem for an element of the universal enveloping algebra of some Lie algebra in the…