## 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

### Keywords:

congruence; affine space
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