Discipline: Mathematics and Statistics
Subcategory: Mathematics and Statistics
Yaniel Rivera - University of Puerto Rico in Cayey
Co-Author(s): Yaziel Rivera, University of Puerto Rico, Cayey, Puerto Rico,Moises Delgado, University of Puerto Rico, Cayey, Puerto Rico,
A multivariate polynomial defined over a field K is absolutely irreducible if it is irreducible over the algebraic closure of the field. Finding criteria to test absolute irreducibility is fundamental for applications in pure and applied mathematics as algebraic geometry, combinatorics, coding theory, cryptography, finite geometry, etc. Some well known mathematical problems that have been solved, or partially solved, by proving absolute irreducibility includes exceptional APN conjecture. There are very few practical criteria for absolute irreducibility known so far. In this research we introduce novel techniques based on the 2 higher-degree terms of polynomials and introduce new absolute irreducibility criteria for multivariate polynomials over finite fields. As a consequence of our techniques, a bound for the number of absolutely irreducible factors of some multivariate polynomials is presented.
Funder Acknowledgement(s): Puerto Rico Louis Stokes Alliance For Minority Participation (PR-LSAMP)
Faculty Advisor: Moises Delgado, email@example.com
Role: Experimented with polynomials of multiple variables for their factorizations and irreducibility using properties of finite fields (using the computer algebra system, SageMath). Investigated new techniques for irreducibility and absolute irreducibility over those polynomials. I’ve managed to find some unique patterns for univariate polynomials that were useful to show irreducibility. Develop a sketch of a proof for why x^n + F(y), where F(y) is any polynomial of the variable y, is absolutely irreducible. I’m now working the general case for F(x) + F(y) been absolutely irreducible.