Multiplicative character sums over two classes of subsets of quadratic extensions of finite fields
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915278937391104 |
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| author | Cheng, Kaimin Winterhof, Arne |
| author_facet | Cheng, Kaimin Winterhof, Arne |
| contents | Let $q$ be a prime power and $r$ a positive even integer. Let $\mathbb{F}_{q}$ be the finite field with $q$ elements and $\mathbb{F}_{q^r}$ be its extension field of degree $r$. Let $χ$ be a nontrivial multiplicative character of $\mathbb{F}_{q^r}$ and $f(X)$ a polynomial over $\mathbb{F}_{q^r}$ with a simple root in $\mathbb{F}_{q^r}$. In this paper, we improve estimates for character sums $\sum\limits_{g \in\mathcal{G}}χ(f(g))$, where $\mathcal{G}$ is either a subset of $\mathbb{F}_{q^r}$ of sparse elements, with respect to some fixed basis of $\mathbb{F}_{q^r}$ which contains a basis of $\mathbb{F}_{q^{r/2}}$, or a subset avoiding affine hyperplanes in general position. While such sums have been previously studied, our approach yields sharper bounds by reducing them to sums over the subfield $\mathbb{F}_{q^{r/2}}$ rather than sums over general linear spaces. These estimates can be used to prove the existence of primitive elements in $\mathcal{G}$ in the standard way. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_14436 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Multiplicative character sums over two classes of subsets of quadratic extensions of finite fields Cheng, Kaimin Winterhof, Arne Number Theory Cryptography and Security 11T23 Let $q$ be a prime power and $r$ a positive even integer. Let $\mathbb{F}_{q}$ be the finite field with $q$ elements and $\mathbb{F}_{q^r}$ be its extension field of degree $r$. Let $χ$ be a nontrivial multiplicative character of $\mathbb{F}_{q^r}$ and $f(X)$ a polynomial over $\mathbb{F}_{q^r}$ with a simple root in $\mathbb{F}_{q^r}$. In this paper, we improve estimates for character sums $\sum\limits_{g \in\mathcal{G}}χ(f(g))$, where $\mathcal{G}$ is either a subset of $\mathbb{F}_{q^r}$ of sparse elements, with respect to some fixed basis of $\mathbb{F}_{q^r}$ which contains a basis of $\mathbb{F}_{q^{r/2}}$, or a subset avoiding affine hyperplanes in general position. While such sums have been previously studied, our approach yields sharper bounds by reducing them to sums over the subfield $\mathbb{F}_{q^{r/2}}$ rather than sums over general linear spaces. These estimates can be used to prove the existence of primitive elements in $\mathcal{G}$ in the standard way. |
| title | Multiplicative character sums over two classes of subsets of quadratic extensions of finite fields |
| topic | Number Theory Cryptography and Security 11T23 |
| url | https://arxiv.org/abs/2502.14436 |