Functiones et Approximatio Commentarii Mathematici

Small solutions of diagonal congruences

Todd Cochrane, Misty Ostergaard, and Craig Spencer

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Abstract

We prove that for $k \geq 2$, $0 <\varepsilon< \frac 1{k(k-1)}$, $n>\frac {k-1}{\varepsilon }$, prime $p> P(\varepsilon, k)$, and integers $c,a_i$, with $p \nmid a_i$, $1 \le i \le n$, there exists a solution $\underline{x}$ to the congruence $$ \sum_{i=1}^n a_ix_i^k \equiv c \mod p $$ in any cube $\mathcal{B}$ of side length $b \ge p^{\frac 1k + \varepsilon}$. Various refinements are given for smaller $n$ and for cubes centered at the origin.

Article information

Source
Funct. Approx. Comment. Math., Volume 56, Number 1 (2017), 39-48.

Dates
First available in Project Euclid: 27 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.facm/1485486016

Digital Object Identifier
doi:10.7169/facm/1587

Mathematical Reviews number (MathSciNet)
MR3629009

Zentralblatt MATH identifier
06864144

Subjects
Primary: 11D79: Congruences in many variables 11D72: Equations in many variables [See also 11P55]
Secondary: 11L03: Trigonometric and exponential sums, general

Keywords
diagonal congruences in many variables exponential sums

Citation

Cochrane, Todd; Ostergaard, Misty; Spencer, Craig. Small solutions of diagonal congruences. Funct. Approx. Comment. Math. 56 (2017), no. 1, 39--48. doi:10.7169/facm/1587. https://projecteuclid.org/euclid.facm/1485486016


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