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A class of algebraic-exponential congruences modulo \(p\). (English) Zbl 1048.11100
Summary: Let \(p\) be a prime number, \(\mathcal J\) a set of consecutive integers, \(\bar \mathbb{F}_p\) the algebraic closure of \(\mathbb{F}_p = \mathbb Z/p\mathbb Z\) and \(\mathfrak C\) an irreducible curve in an affine space \(\mathbb A^r(\bar \mathbb{F}_p)\), defined over \(\bar \mathbb{F}_p\). We prove a lower bound for the number of \(r\)-tuples \((x,y_1,\dots ,y_{r-1})\) with \(x\in \mathcal J,\, y_1,\dots ,y_{r-1}\in \{0,1,\cdots ,p-1\}\) for which \((x,y^x_1,\dots ,y^x_{r-1}) (\text{mod}\, p)\) belongs to \(C\,(\bar \mathbb{F}_p)\).
MSC:
11T99 Finite fields and commutative rings (number-theoretic aspects)
11A07 Congruences; primitive roots; residue systems
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