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On the sumsets of polynomial units in a finite commutative ring

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Abstract

Let \(k\) be an integer with \(k\ge 2\). Let \(R\) be a finite commutative ring with zero element \(0_R\) and identity element \(1_R\neq 0_R\) and let \(R^*\) be the multiplicative group of units of \(R\). Let \(f(x)\in R[x]\) be a non-constant polynomial. An element \(u\in R\) is called an \(f\)-unit if \(f(u)\in R^*\). An \(f\)-unit is called an exceptional unit when \(f(x)=x(1_R-x)\). In this paper, we obtain an exact formula for the number of representations of any element of R as the sum of k f-units of R. Furthermore, by using the technique of exponential sums, we deduce a more explicit formula for the case when f(x) is linear or quadratic. Our results generalize Miguel's theorem from exceptional unit to general f-unit and the Zhao–Hong–Zhu theorem from the ring of residue classes to the general finite commutative ring.

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Correspondence to S. A. Hong.

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Y. L. Feng was supported partially by National Science Foundation of China, Grant #12171332.

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Feng, Y.L., Hong, S.A. On the sumsets of polynomial units in a finite commutative ring. Acta Math. Hungar. 167, 180–191 (2022). https://doi.org/10.1007/s10474-022-01238-x

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