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### Session S26 - Finite fields and applications

Friday, July 16, 14:00 ~ 14:20 UTC-3

## On diagonal equations over finite fields via walks in NEPS of graphs

### Denis Videla

#### Universidad Nacional de Córdoba, Argentina   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloakdd819572c5f6fe383663487f7a39b7b8').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addydd819572c5f6fe383663487f7a39b7b8 = 'd&#101;n&#105;sv458' + '&#64;'; addydd819572c5f6fe383663487f7a39b7b8 = addydd819572c5f6fe383663487f7a39b7b8 + 'gm&#97;&#105;l' + '&#46;' + 'c&#111;m'; var addy_textdd819572c5f6fe383663487f7a39b7b8 = 'd&#101;n&#105;sv458' + '&#64;' + 'gm&#97;&#105;l' + '&#46;' + 'c&#111;m';document.getElementById('cloakdd819572c5f6fe383663487f7a39b7b8').innerHTML += '<a ' + path + '\'' + prefix + ':' + addydd819572c5f6fe383663487f7a39b7b8 + '\'>'+addy_textdd819572c5f6fe383663487f7a39b7b8+'<\/a>';

We obtain an explicit combinatorial formula for the number of solutions $(x_1,\ldots,x_r)\in \mathbb{F}_{p^{ab}}$ to the diagonal equation $x_{1}^k+\cdots+x_{r}^k=\alpha$ over the finite field $\mathbb{F}_{p^{ab}}$, with $k=\frac{p^{ab}-1}{b(p^a-1)}$ and $b>1$ by using the number of $r$-walks in NEPS of complete graphs. This talk is based on a recent accepted article.

\textsc{Denis E.\@ Videla}. \textit{On diagonal equations over finite fields via walks in NEPS of graphs}. Finite Fields Appl. (2021), accepted, https://arxiv.org/abs/1907.03145