On a quadratic Waring's problem with congruence conditions

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 certain congruence condition. We prove that the growth of $g_\Delta(n)$ is at most an exponential of $\sqrt{n}$, 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 $4$.