## A Polynomial Sieve and Sums of Deligne Type

Publication
, Journal Article

Bonolis, D

Published in: International Mathematics Research Notices

January 1, 2021

Let $f\in \mathbb{Z}[T]$ be any polynomial of degree $d>1$ and $F\in \mathbb{Z}[X-{0},..,X-{n}]$ an irreducible homogeneous polynomial of degree $e>1$ such that the projective hypersurface $V(F)$ is smooth. In this paper we present a new bound for $N(f,F,B):=|\{\textbf{x}\in \mathbb{Z}^{n+1}:\max-{0\leq i\leq n}|x-{i}|\leq B,\exists t\in \mathbb{Z}\textrm{ such that}\ f(t)=F(\textbf{x})\}|.$ To do this, we introduce a generalization of the power sieve [14, 28] and we extend two results by Deligne and Katz on estimates for additive and multiplicative characters in many variables.

### Duke Scholars

## Published In

International Mathematics Research Notices

## DOI

## EISSN

1687-0247

## ISSN

1073-7928

## Publication Date

January 1, 2021

## Volume

2021

## Issue

2

## Start / End Page

1096 / 1137

## Related Subject Headings

- General Mathematics
- 4902 Mathematical physics
- 0101 Pure Mathematics

### Citation

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Chicago

ICMJE

MLA

NLM

Bonolis, D. (2021). A Polynomial Sieve and Sums of Deligne Type.

*International Mathematics Research Notices*,*2021*(2), 1096–1137. https://doi.org/10.1093/imrn/rnz245Bonolis, D. “A Polynomial Sieve and Sums of Deligne Type.”

*International Mathematics Research Notices*2021, no. 2 (January 1, 2021): 1096–1137. https://doi.org/10.1093/imrn/rnz245.Bonolis D. A Polynomial Sieve and Sums of Deligne Type. International Mathematics Research Notices. 2021 Jan 1;2021(2):1096–137.

Bonolis, D. “A Polynomial Sieve and Sums of Deligne Type.”

*International Mathematics Research Notices*, vol. 2021, no. 2, Jan. 2021, pp. 1096–137.*Scopus*, doi:10.1093/imrn/rnz245.Bonolis D. A Polynomial Sieve and Sums of Deligne Type. International Mathematics Research Notices. 2021 Jan 1;2021(2):1096–1137.

## Published In

International Mathematics Research Notices

## DOI

## EISSN

1687-0247

## ISSN

1073-7928

## Publication Date

January 1, 2021

## Volume

2021

## Issue

2

## Start / End Page

1096 / 1137

## Related Subject Headings

- General Mathematics
- 4902 Mathematical physics
- 0101 Pure Mathematics