Abstract:We study sums of higher even powers in the coordinate rings of singular planar curves $x^M=y^m$ for coprime positive integers $m<M$. We then show that the $2s$-Pythagoras number of real algebras of the form $\mathbb{R}[x,y,\sqrt{f_1},\sqrt{f_2},\dots, \sqrt{f_n}]$ are infinite, under some mild assumptions on the polynomials $f_1,f_2,\dots, f_n \in \mathbb{R}[x,y]$. We prove that all of the higher even Pythagoras numbers are finite for the ring of $0$-regulous functions on a $0$-regulous variety. We then show that the codimension of the bad set of order $2n$, for $n>1$, can be of codimension $2$, contrary to the quadratic case.
From: Tomasz Kowalczyk [view email]
[v1]
Sun, 21 Jun 2026 09:15:40 UTC (19 KB)
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