## On the polynomial Szemerédi theorem in finite fields.(English)Zbl 1473.11032

Summary: Let $$P_1,\ldots,P_m\in\mathbb{Z}[y]$$ be any linearly independent polynomials with zero constant term. We show that there exists $$\gamma> 0$$ such that any subset of $$\mathbb{F}_q$$ of size at least $$q^{1-\gamma}$$ contains a nontrivial polynomial progression $$x,x+P_{1}(y),\ldots,x+P_{m}(y)$$, provided that the characteristic of $$\mathbb{F}_q$$ is large enough.

### MSC:

 11B30 Arithmetic combinatorics; higher degree uniformity 11B25 Arithmetic progressions

### Keywords:

Szemerédi’s theorem; finite fields
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