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Results in Mathematics

, 74:65 | Cite as

Irreducibility Criteria for the Sum of Two Relatively Prime Multivariate Polynomials

  • Anca Iuliana Bonciocat
  • Nicolae Ciprian BonciocatEmail author
Article
  • 24 Downloads

Abstract

We provide irreducibility criteria for multivariate polynomials over a field K, of the form \(f+p^{k}g\), where \(f,g\in K[X_{1},\dots ,X_{r}]\), \(\deg _{r}f<\deg _{r}g\), \(p\in K[X_{1},\dots ,X_{r-1}]\) is irreducible over \(K(X_{1},\dots ,X_{r-2})\), and \(k\ge 1\) is an integer prime to \(\deg _{r}g\). More precisely, we prove that if f and g regarded as polynomials in \(X_{r}\) with coefficients in \(K[X_{1},\dots ,X_{r-1}]\) are relatively prime over \(K(X_{1},\dots ,X_{r-1})\), k is prime to \(\deg _{r}g\), and \(\deg _{r-1}p\) is sufficiently large, then the polynomial \(f+p^{k}g\) is irreducible over \(K(X_{1},\dots ,X_{r-1})\). This complements some previous results where k was assumed to be prime to \(\deg _{r}g-\deg _{r}f\).

Keywords

irreducible polynomials Newton polygon resultant 

Mathematics Subject Classification

Primary 11R09 Secondary 11C08 

Notes

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Simion Stoilow Institute of Mathematics of the Romanian AcademyBucharestRomania
  2. 2.Simion Stoilow Institute of Mathematics of the Romanian AcademyBucharestRomania

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