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On a quadratic Waring's problem with congruence conditions

Volume 194 / 2020

Daejun Kim Acta Arithmetica 194 (2020), 73-97 MSC: Primary 11E12, 11E25. DOI: 10.4064/aa190211-17-10 Published online: 28 February 2020


For each positive integer $n$, let $g_\Delta (n)$ be the smallest positive integer $g$ such that every complete quadratic polynomial in $n$ variables which can be represented by a sum of odd squares is represented by a sum of at most $g$ odd squares. In this paper, we analyze $g_\Delta (n)$ by studying representations of integral quadratic forms by sums of squares with a certain congruence condition. We prove that the growth of $g_\Delta (n)$ is at most exponential in $\sqrt {n}$, which is the same as the best known upper bound on the $g$-invariants of the original quadratic Waring’s problem. We also determine the exact value of $g_\Delta (n)$ for each positive integer less than or equal to $4$.


  • Daejun KimDepartment of Mathematical Sciences
    Seoul National University
    Seoul 151-747, Korea

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