Problems Around Polynomials — Problem 3

Definitions and mathematical statement

Fix integers $k,\ell\geq1$. Consider polynomials $P\in \mathbb R[x_1,\ldots,x_\ell]$ of total degree exactly $2k$ admitting a representation \[ P=Q_1^2+\cdots+Q_s^2,\qquad s\geq1,\quad Q_j\in\mathbb R[x_1,\ldots,x_\ell],\quad \deg Q_j\leq k. \] The number $s$ is unrestricted and may depend on $P$. Such a polynomial is nonnegative on real space, and its real zero set is exactly the common real zero set of all the $Q_j$.

A zero $a$ is isolated if a neighborhood of $a$ contains no other common zero. Let $Z_{\mathrm{iso}}(P)$ count these distinct real points, and put \[ N_{\mathrm{SOS}}(k,\ell) =\sup_P Z_{\mathrm{iso}}(P). \] Positive-dimensional components may coexist with the isolated zeros; they do not contribute to the count. Neither complex zeros nor projective points at infinity are counted.

Determine this extremal function for arbitrary $k$ and $\ell$, including sharp upper bounds and constructions attaining them. A basic lower bound is $k^\ell$: take one summand $Q_j=\prod_{s=1}^{k}(x_j-s)$ for each coordinate $j$. The question concerns all sums of squares, including representations with more summands than variables, rather than only square systems of $\ell$ equations. Consequently a count obtained for a generic complete intersection does not by itself establish an upper bound for this class. This is the general extremal problem, separate from any proposed closed formula for its answer.

Short English statement

How many isolated common real zeros can a sum of squares of bounded-degree polynomials have?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2200006 (AMR-021-0006), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Shapiro - Problems Around Polynomials: The Good, The Bad and The Ugly (2015), source-order item 6 (Problem 3 as printed). Original source indexed by AMR-021-0006. https://arxiv.org/abs/1503.05295.