Büchi’s Problem in Modular Arithmetic for Arbitrary Quadratic Polynomials
Canadian mathematical bulletin, Tome 62 (2019) no. 4, pp. 876-885
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Given a prime $p\geqslant 5$ and an integer $s\geqslant 1$, we show that there exists an integer $M$ such that for any quadratic polynomial $f$ with coefficients in the ring of integers modulo $p^{s}$, such that $f$ is not a square, if a sequence $(f(1),\ldots ,f(N))$ is a sequence of squares, then $N$ is at most $M$. We also provide some explicit formulas for the optimal $M$.
Sáez, Pablo; Vidaux, Xavier; Vsemirnov, Maxim. Büchi’s Problem in Modular Arithmetic for Arbitrary Quadratic Polynomials. Canadian mathematical bulletin, Tome 62 (2019) no. 4, pp. 876-885. doi: 10.4153/S0008439519000225
@article{10_4153_S0008439519000225,
author = {S\'aez, Pablo and Vidaux, Xavier and Vsemirnov, Maxim},
title = {B\"uchi{\textquoteright}s {Problem} in {Modular} {Arithmetic} for {Arbitrary} {Quadratic} {Polynomials}},
journal = {Canadian mathematical bulletin},
pages = {876--885},
year = {2019},
volume = {62},
number = {4},
doi = {10.4153/S0008439519000225},
url = {http://geodesic.mathdoc.fr/articles/10.4153/S0008439519000225/}
}
TY - JOUR AU - Sáez, Pablo AU - Vidaux, Xavier AU - Vsemirnov, Maxim TI - Büchi’s Problem in Modular Arithmetic for Arbitrary Quadratic Polynomials JO - Canadian mathematical bulletin PY - 2019 SP - 876 EP - 885 VL - 62 IS - 4 UR - http://geodesic.mathdoc.fr/articles/10.4153/S0008439519000225/ DO - 10.4153/S0008439519000225 ID - 10_4153_S0008439519000225 ER -
%0 Journal Article %A Sáez, Pablo %A Vidaux, Xavier %A Vsemirnov, Maxim %T Büchi’s Problem in Modular Arithmetic for Arbitrary Quadratic Polynomials %J Canadian mathematical bulletin %D 2019 %P 876-885 %V 62 %N 4 %U http://geodesic.mathdoc.fr/articles/10.4153/S0008439519000225/ %R 10.4153/S0008439519000225 %F 10_4153_S0008439519000225
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