Primitive zeros of quadratic forms mod $p^2$

Authors

  • Ali H. Hakami

DOI:

https://doi.org/10.5556/j.tkjm.46.2015.1745

Keywords:

Quadratic forms, congruences, small solutions

Abstract

Let $Q({\bf{x}}) = Q(x_1 ,x_2 ,\ldots,x_n )$ be a quadratic form with integer coefficients, $p$ be an odd prime and $\left\| \bf{x} \right\| = \max _i \left| {x_i } \right|.$ A solution of the congruence $Q({\mathbf{x}}) \equiv {\mathbf{0}}\;(\bmod\; p^2 )$ is said to be a primitive solution if $p\nmid x_i $ for some $i$. In this paper, we seek to obtain primitive solutions of this congruence in small rectangular boxes of the type $ \mathcal{B} = \{ {\mathbf{x}} \in \mathbb{Z}^n : |x_i| \le M_i ,\;1 \leqslant i \leqslant n\} $ where for $1 \le i \le l$ we have $M_i \le p$, while for $i>l$ we have $M_i>p$. In particular, we show that if $n \ge 4$, $n$ even, $l \le \frac n2-2$, and $Q$ is nonsingular $\pmod p$, then there exists a primitive solution with $x_i = 0$, $1 \le i \le l$, and $|x_i| \le 2^{\frac {4n+3}{n-l}} p^{\frac n{n-l}} +1$, for $l<i \le n$.

Author Biography

Ali H. Hakami

Department ofMathematics, Jazan University, P.O.Box 277, Jazan, Postal Code: 45142, Saudi Arabia.

References

T. Cochrane, Small zeros of quadratic forms modulo $p$, J. Number Theory, 33(1989), 286--292.

T. Cochrane, Small zeros of quadratic forms modulo $p$, II, Proceedings of the Illinois Number Theory Conference, (1989), Birkhauser, Boston (1990), 91--94.

T. Cochrane, Small zeros of quadratic forms modulo $p$, III, J. Number Theory, 33 (1991), 92--99.

T. Cochrane, Small zeros of quadratic congruences modulo $pq$, Mathematika, 37(1990), 261--272.

T. Cochrane, Small zeros of quadratic congruences modulo $pq$, II, J. Number Theory 50(1995), 299--308.

T. Cochrane and A. Hakami, Small zeros of quadratic congruences modulo $p^2$, Proceedings of the American Mathematical Society, 140 (2012), 4041--4052.

A. Hakami, Small zeros of quadratic congruences to a prime power modulus, PhD thesis, Kansas State University, 2009.

A. Hakami, Small zeros of quadratic forms modulo $p^2$, JP J. Algebra, Number Theory and Applications, 17(2011), 141--162.

A. Hakami, Small zeros of quadratic forms modulo $p^3$, Advances and Applications in Mathematical Sciences, 9(2011), 47--69.

A. Hakami, Small primitive zeros of quadratic forms modulo $p^m$, The Ramanujan J (2014), DOI 10.1007/s11139--014--9614--3.

A. Hakami, On Cochrane's estimate for small zeros of quadratic forms modulo $p$, Far East J. Math. Sciences, 50(2011), 151--157.

A. Hakami, An upper bound for the number of integral solutions of quadratic forms modulo $p$, J. Algebra, Number Theory: Advances and Applications, 6 (2011), 1--17.

A. Hakami, Weighted quadratic partitions $(mod ;p^m),$ , A new formula and new demonstration, Tamaking J. Math., 43(2012), 11--19.

L. Carlitz, Weighted quadratic partitions $(mod ;p^r),$, Math Zeitschr. Bd, 59(1953), 40--46.

D. R. Heath--Brown, Small solutions of quadratic congruences, Glasgow Math. J, 27 (1985), 87--93.

D.R. Heath--Brown, Small solutions of quadratic congruences II, Mathematika, 38(1991), 264--284.

A. Schinzel, H.P. Schlickewei and W.M. Schmidt, Small solutions of quadratic congruences and small fractional parts of quadratic forms, Acta Arithmetica, 37 (1980), 241--248.

Y. Wang, On small zeros of quadratic forms over finite fields, Algebraic structures and number theory (Hong Kong, 1988), 269----274, World Sci. Publ., Teaneck, NJ, 1990.

Y. Wang, On small zeros of quadratic forms over finite fields, J. Number Theory, 31(1989), 272--284.

Y. Wang, On small zeros of quadratic forms over finite fields II, A Chinese summary appears in Acta Math. Sinica, 37(1994), 719--720. Acta Math. Sinica (N.S.), 9(1993), 382--389.

Downloads

Published

2015-09-30

How to Cite

Hakami, A. H. (2015). Primitive zeros of quadratic forms mod $p^2$. Tamkang Journal of Mathematics, 46(3), 349–364. https://doi.org/10.5556/j.tkjm.46.2015.1745

Issue

Section

Papers