arXiv subjects

Nicolae Ciprian Bonciocat

Publications and source records attributed to Nicolae Ciprian Bonciocat.

8 recordsLinked to original sources

arXiv2025

Logarithmic Newton polygons and polytopes, and the factorization of Dirichlet polynomials

To study a Dirichlet polynomial $f(s)=\frac{a_{m}}{m^{s}}+\cdots +\frac{a_{n}}{n^{s}}$ by regarding it as a multivariate polynomial via the canonical map $ϕ$ sending $p_i^{-s}$ to an indeterminate $X_i$, with $p_i$ the $i$th prime number, requires knowing the prime factorizations of all the integers in the support of $f$. We devise several methods to study the factorization of Dirichlet polynomials over unique factorization domains that circumvent the use of $ϕ$, and obtain irreducibility criteria that are analogous to the classical results of Schönemann, Eisenstein, Dumas, Stäckel, Ore and Weisner for polynomials, and to more recent results of Filaseta and Cavachi. Some of the proofs rely on logarithmic versions of the classical Newton polygons. Criteria that use two or more $p$-adic valuations by combining information from different logarithmic Newton polygons of $f$, as well as irreducibility conditions for Dirichlet polynomials that assume a prime or a prime power value are also obtained. We also find excluding intervals for the relative degrees of the factors of a Dirichlet polynomial, and upper bounds for the multiplicities of the irreducible factors, in particular square-free criteria, that use no derivatives. Criteria of absolute irreducibility analogous to results of Ostrowski, Gao and Stepanov-Schmidt are finally provided in the multivariate case by using logarithmic Newton polytopes and logarithmic upper Newton polygons.

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arXiv2025

Irreducibility criteria for pairs of polynomials whose resultant is a prime number

We obtain various irreducibility criteria for pairs of polynomials $(f(X),g(X))$ with integer coefficients whose resultant $Res(f,g)$ is a prime number, or is divisible by a sufficiently large prime number, and also for some of their linear combinations $Mf(X)+Ng(X)$ with integer scalars $M$ and $N$. In particular, we find irreducibility conditions for polynomials with coefficients obtained by representing primes by certain quadratic forms. The irreducibility criteria will appear as corollaries of more general results providing upper bounds for the number of irreducible factors of each one of $f$ and $g$, counting multiplicities, that depend on the prime factorization of $Res(f,g)$, and on the distances between the roots of $f$ and those of $g$. Similar results will be also obtained for pairs of bivariate polynomials $(f(X,Y),g(X,Y))$ over an arbitrary field $K$, using information on the canonical decomposition of their resultant $Res_Y(f,g)$, and on the location of their roots in an algebraic closure of $K(X)$, studied in a non-Archimedean setting.

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arXiv2024

Some factorization results for bivariate polynomials

We provide upper bounds on the total number of irreducible factors, and in particular irreducibility criteria for some classes of bivariate polynomials $f(x,y)$ over an arbitrary field $\mathbb{K}$. Our results rely on information on the degrees of the coefficients of $f$, and on information on the factorization of the constant term and of the leading coefficient of $f$, viewed as a polynomial in $y$ with coefficients in $\mathbb{K}[x]$. In particular, we provide a generalization of the bivariate version of Perron's irreducibility criterion, and similar results for polynomials in an arbitrary number of indeterminates. The proofs use non-Archimedean absolute values, that are suitable for finding information on the location of the roots of $f$ in an algebraic closure of $\mathbb{K}(x)$.

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arXiv2023

Apollonius circles and the number of irreducible factors of polynomials

We provide upper bounds for the sum of the multiplicities of the non-constant irreducible factors that appear in the canonical decomposition of a polynomial $f(X)\in\mathbb{Z}[X]$, in case all the roots of $f$ lie inside an Apollonius circle associated to two points on the real axis with integer abscissae $a$ and $b$, with ratio of the distances to these points depending on the admissible divisors of $f(a)$ and $f(b)$. In particular, we obtain such upper bounds for the case where $f(a)$ and $f(b)$ have few prime factors, and $f$ is an Eneström-Kakeya polynomial, or a Littlewood polynomial, or has a large leading coefficient. Similar results are also obtained for multivariate polynomials over arbitrary fields, in a non-Archimedean setting.

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arXiv2021

Zero-free angular sectors and lens-shaped regions for polynomials, with applications to irreducibility

For a real polynomial $f$ we present explicit zero-free angular sectors in the complex plane, symmetric with respect to the real axis, with angles depending only on the degree of $f$, and vertices expressed in terms of the coefficients of $f$. We also describe zero-free lens-shaped regions for $f$ that are associated to the zero-free sectors of the reciprocal of $f$. As an application, we use these zero-free regions to obtain irreducibility criteria for polynomials with integer coefficients that take a prime or a prime power value.

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arXiv2021

Apollonius circles and irreducibility criteria for polynomials

We prove the irreducibility of integer polynomials $f(X)$ whose roots lie inside an Apollonius circle associated to two points on the real axis with integer abscisae $a$ and $b$, with ratio of the distances to these points depending on the canonical decomposition of $f(a)$ and $f(b)$. In particular, we obtain irreducibility criteria for the case where $f(a)$ and $f(b)$ have few prime factors, and $f$ is either an Eneström-Kakeya polynomial, or has a large leading coefficient. Analogous results are also provided for multivariate polynomials over arbitrary fields, in a non-Archimedean setting.

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arXiv2020

There is no Diophantine $D(-1)$--quadruple

A set of positive integers with the property that the product of any two of them is the successor of a perfect square is called Diophantine $D(-1)$--set. Such objects are usually studied via a system of generalized Pell equations naturally attached to the set under scrutiny. In this paper, an innovative technique is introduced in the study of Diophantine $D(-1)$--quadruples. The main novelty is the uncovering of a quadratic equation relating various parameters describing a hypothetical $D(-1)$--quadruple with integer entries. In combination with extensive computations, this idea leads to the confirmation of the conjecture according to which there is no Diophantine $D(-1)$--quadruple.

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arXiv2014

Schönemann-Eisenstein-Dumas-type irreducibility conditions that use arbitrarily many prime numbers

The famous irreducibility criteria of Schönemann-Eisenstein and Dumas rely on information on the divisibility of the coefficients of a polynomial by a single prime number. In this paper we provide several irreducibility criteria of Schönemann-Eisenstein-Dumas-type for polynomials with integer coefficients, criteria that are given by some divisibility conditions for their coefficients with respect to arbitrarily many prime numbers. A special attention will be paid to those irreducibility criteria that require information on the divisibility of the coefficients by two distinct prime numbers.

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