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Integer Linear Programming applied to determining monic hyperbolic irreducible polynomials with integer coefficients and span less than 4

Souad El Otmani, Armand Maul, Georges Rhin, Jean-Marc Sac-Épée (2013)

Journal de Théorie des Nombres de Bordeaux

In this work, we propose a new method to find monic irreducible polynomials with integer coefficients, only real roots, and span less than 4. The main idea is to reduce the search of such polynomials to the solution of Integer Linear Programming problems. In this frame, the coefficients of the polynomials we are looking for are the integer unknowns. We give inequality constraints specified by the properties that the polynomials should have, such as the typical distribution of their roots. These...

Integer-valued polynomials on algebras: a survey

Sophie Frisch (2010)

Actes des rencontres du CIRM

We compare several different concepts of integer-valued polynomials on algebras and collect the few results and many open questions to be found in the literature.

Inverse zero-sum problems in finite Abelian p-groups

Benjamin Girard (2010)

Colloquium Mathematicae

We study the minimal number of elements of maximal order occurring in a zero-sumfree sequence over a finite Abelian p-group. For this purpose, and in the general context of finite Abelian groups, we introduce a new number, for which lower and upper bounds are proved in the case of finite Abelian p-groups. Among other consequences, our method implies that, if we denote by exp(G) the exponent of the finite Abelian p-group G considered, every zero-sumfree sequence S with maximal possible length over...

Invertible ideals and Gaussian semirings

Shaban Ghalandarzadeh, Peyman Nasehpour, Rafieh Razavi (2017)

Archivum Mathematicum

In the first section, we introduce the notions of fractional and invertible ideals of semirings and characterize invertible ideals of a semidomain. In section two, we define Prüfer semirings and characterize them in terms of valuation semirings. In this section, we also characterize Prüfer semirings in terms of some identities over its ideals such as ( I + J ) ( I ∩ J ) = I J for all ideals I , J of S . In the third section, we give a semiring version for the Gilmer-Tsang Theorem, which states that for a suitable family...

Irreducible Jacobian derivations in positive characteristic

Piotr Jędrzejewicz (2014)

Open Mathematics

We prove that an irreducible polynomial derivation in positive characteristic is a Jacobian derivation if and only if there exists an (n-1)-element p-basis of its ring of constants. In the case of two variables we characterize these derivations in terms of their divergence and some nontrivial constants.

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