On Alternatives of Polynomial Congruences

Volume 52 / 2004

Mariusz Skałba Bulletin Polish Acad. Sci. Math. 52 (2004), 123-132 MSC: 11R20, 11A15. DOI: 10.4064/ba52-2-3

Abstract

What should be assumed about the integral polynomials $f_{1}(x),\ldots,f_{k}(x)$ in order that the solvability of the congruence $f_{1}(x)f_{2}(x)\cdots f_{k}(x)\equiv 0\pmod{p}$ for sufficiently large primes $p$ implies the solvability of the equation $f_{1}(x)f_{2}(x)\cdots f_{k}(x)=0$ in integers $x$? We provide some explicit characterizations for the cases when $f_j(x)$ are binomials or have cyclic splitting fields.

Authors

  • Mariusz SkałbaInstitute of Mathematics
    Polish Academy of Sciences
    P.O. Box 21
    00-956 Warszawa, Poland
    e-mail

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