Skew fields of noncommutative rational functions (preliminary version)
Skolem–Mahler–Lech type theorems and Picard–Vessiot theory
We show that three problems involving linear difference equations with rational function coefficients are essentially equivalent. The first problem is the generalization of the classical Skolem–Mahler–Lech theorem to rational function coefficients. The second problem is whether or not for a given linear difference equation there exists a Picard–Vessiot extension inside the ring of sequences. The third problem is a certain special case of the dynamical Mordell–Lang conjecture. This allows us to deduce...
Slope filtration of quasi-unipotent overconvergent -isocrystals
We study local properties of quasi-unipotent overconvergent -isocrystals on a curve over a perfect field of positive characteristic . For a --module over the Robba ring , we define the slope filtration for Frobenius structures. We prove that an overconvergent -isocrystal is quasi-unipotent if and only if it has the slope filtration for Frobenius structures locally at every point on the complement of the curve.
Slope filtrations revisited.
Small Galois groups that encode valuations
Small maximal pro-p Galois groups.
Smith forms of palindromic matrix polynomials.
Smith normal form of a matrix of generalized polynomials with rational exponents
It is proved that generalized polynomials with rational exponents over a commutative field form an elementary divisor ring; an algorithm for computing the Smith normal form is derived and implemented.
Solution d'un problème d'Erdös, Gillman et Henriksen et application à l'étude des homomorphismes de C-(K).
Solutions rationnelles d'équations différentielles. Application à un problème de transcendance
Solvable-by-finite groups as differential Galois groups
Solved and unsolved problems οn polynomials
Solving algebraic equations in roots of unity.
Solving Equations, an Elegant Legacy
Solving quadratic equations over polynomial rings of characteristic two.
We are concerned with solving polynomial equations over rings. More precisely, given a commutative domain A with 1 and a polynomial equation antn + ...+ a0 = 0 with coefficients ai in A, our problem is to find its roots in A.We show that when A = B[x] is a polynomial ring, our problem can be reduced to solving a finite sequence of polynomial equations over B. As an application of this reduction, we obtain a finite algorithm for solving a polynomial equation over A when A is F[x1, ..., xN] or F(x1,...
Solving the quintic by iteration in three dimensions.
Some algebraic aspects of quadratic forms over fields of characteristic two.
Some Algebraic Properties of Polynomial Rings
In this article we extend the algebraic theory of polynomial rings, formalized in Mizar [1], based on [2], [3]. After introducing constant and monic polynomials we present the canonical embedding of R into R[X] and deal with both unit and irreducible elements. We also define polynomial GCDs and show that for fields F and irreducible polynomials p the field F[X]/ is isomorphic to the field of polynomials with degree smaller than the one of p.
Some basic facts in algebraic geometry on a non algebraically closed field
Some classes of irreducible polynomials