Equations in one variable over function fields
Errata to "Average orders of multiplicative arithmetical functions of integer matrices" (Acta Arith. 66 (1994), 45-62)
Étude bibliographique. Sur la théorie des nombres
Evaluation of divisor functions of matrices
1. Introduction. The study of divisor functions of matrices arose legitimately in the context of arithmetic of matrices, and the question of the number of (possibly weighted) inequivalent factorizations of an integer matrix was asked. However, till now only partial answers were available. Nanda [6] evaluated the case of prime matrices and Narang [7] gave an evaluation for 2×2 matrices. We obtained a recursion in the size of the matrices and the weights of the divisors [1,2] which helped us obtain...
Evaluations of some determinants of matrices related to the Pascal triangle.
Exponential polynomial inequalities and monomial sum inequalities in -Newton sequences
We consider inequalities between sums of monomials that hold for all p-Newton sequences. This continues recent work in which inequalities between sums of two, two-term monomials were combinatorially characterized (via the indices involved). Our focus is on the case of sums of three, two-term monomials, but this is very much more complicated. We develop and use a theory of exponential polynomial inequalities to give a sufficient condition for general monomial sum inequalities, and use the sufficient...
Exponential Riordan arrays and permutation enumeration.
Exposé des principes élémentaires de la théorie des déterminants, à l'usage des élèves de mathématiques spéciales
Extended GCD and Hermite normal form algorithms via lattice basis reduction.
Extensions of the Bloch–Pólya theorem on the number of real zeros of polynomials
We prove that there are absolute constants and such that for everythere aresuch thathas at least distinct sign changes in . This improves and extends earlier results of Bloch and Pólya.
Factoring Polynomials with Rational Coefficients.
Factorisation in fractional powers
Factorization of matrices associated with classes of arithmetical functions
Let f be an arithmetical function. A set S = x₁,..., xₙ of n distinct positive integers is called multiple closed if y ∈ S whenever x|y|lcm(S) for any x ∈ S, where lcm(S) is the least common multiple of all elements in S. We show that for any multiple closed set S and for any divisor chain S (i.e. x₁|...|xₙ), if f is a completely multiplicative function such that (f*μ)(d) is a nonzero integer whenever d|lcm(S), then the matrix having f evaluated at the greatest common divisor of and as its...
Fermat's Equation in Matrices
The Fermat equation is solved in integral two by two matrices of determinant one as well as in finite order integral three by three matrices.
Fermat's theorem for matrices revisited
Fifteen problems in number theory.
Finding integers k for which a given Diophantine equation has no solution in kth powers of integers
Finding the roots of polynomial equations: an algorithm with linear command.
We show how an old principle, due to Walsh (1922), can be used in order to construct an algorithm which finds the roots of polynomials with complex coefficients. This algorithm uses a linear command. From the very first step, the zero is located inside a disk, so several zeros can be searched at the same time.
Finiteness of a class of Rabinowitsch polynomials
We prove that there are only finitely many positive integers such that there is some integer such that is 1 or a prime for all , thus solving a problem of Byeon and Stark.
Fixpunktmannigfaltigkeiten symplektischer Matrizen