**Graduate #56**

**Discipline:**Mathematics and Statistics

**Subcategory:**Mathematics and Statistics

**Session:**1

**Room:**Park Tower 8211

**Carlos Seda**

**- University of Puerto Rico, Rio Piedras**

**Co-Author(s):**Ivelisse Rubio, University of Puerto Rico, Rio Piedras

One of the fundamental problems in mathematics is whether or not it is possible to obtain a solution for a system of polynomial equations. In ”An Improvement of a Theorem of Carlitz” [2], Castro, Moreno and Rubio generalize the results of Moreno-Moreno’s Theorem [3] that gives a sufficient condition for a power of prime to divide the number of common zeros of a system of multivariate polynomials. This generalization regarded the coefficients of said polynomials to be uni-variate polynomials over a finite field instead of plain elements of the finite field. Consequently, this result led to improve a theorem of Carlitz [1], for the estimation of the number of variables needed so that a system of polynomial equations with coefficients that are uni-variate polynomials over a finite field can have non-trivial solutions. It is suggested, in [2], that these results can be further generalized to systems of polynomials with multivariate polynomials over finite fields as coefficients. In this work, we will generalize the results of Castro, Moreno and Rubio to systems of polynomials with multivariate polynomials as coefficients. Our plan is to follow the strategy used in [2] of using the reduction to the ground field technique.

References

[1] L. Carlitz, Some applications of a theorem of Chevalley. Duke Math. J.18 (1951), no. 4, 811-819.

[2] F. Castro, O. Moreno, I. Rubio. An Improvement of a Theorem of Carlitz. Submitted to JPAA. (2017)

[3] O. Moreno, C.J. Moreno, Improvements of the Chevalley-Warning and the Ax-Katz Theorems. American Journal of Mathematics, Vol. 117, No. 1 (Feb., 1995), pp. 241-244.

**Funder Acknowledgement(s):** NSF
; PR-LSAMP
; Bridge to the Doctorate Fellowship

**Faculty Advisor: **Ivelisse Rubio,
iverubio@gmail.com

**Role:** In this research, I will be studying the preliminary mathematical background that is needed in order to provide the generalization of the results in ''An Improvement of a Theorem of Carlitz'' by Castro, Moreno and Rubio.