Problems Around Polynomials — Problem 2

Definitions and mathematical statement

For integers $k,\ell\geq1$, let $\mathcal N_{k,\ell}$ be the set of real polynomials in $\ell$ variables of total degree exactly $2k$ that are nonnegative at every point of $\mathbb R^\ell$. For $P\in\mathcal N_{k,\ell}$, a point $a\in\mathbb R^\ell$ is an isolated zero if $P(a)=0$ and some Euclidean ball centered at $a$ contains no other zero of $P$. Count distinct points, without algebraic multiplicities.

Define \[ Z_{\mathrm{iso}}(P)=\#\{a:a\text{ is an isolated real zero of }P\}, \qquad N(k,\ell)=\sup_{P\in\mathcal N_{k,\ell}}Z_{\mathrm{iso}}(P). \] Other components of the zero set are allowed; only isolated points contribute to this count. Zeros at infinity of a homogenization are not included. Coefficients are arbitrary real numbers, with no bound on their size and no sum-of-squares assumption.

Determine $N(k,\ell)$, or obtain sharp asymptotics as $k\to\infty$ for each fixed $\ell$. Sharpness requires examples as well as an upper bound valid for every admissible polynomial. The scale is already nontrivial: the polynomial \[ P(x_1,\ldots,x_\ell)=\sum_{j=1}^{\ell} \left(\prod_{s=1}^{k}(x_j-s)\right)^2 \] has precisely the $k^\ell$ points of $\{1,\ldots,k\}^{\ell}$ as zeros. This construction supplies a lower bound, but is not a conjectural formula imposed on the unrestricted nonnegative class.

Short English statement

Determine the largest possible number of isolated real zeros of a nonnegative polynomial of prescribed degree and number of variables.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2200005 (AMR-021-0005), 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 5 (Problem 2 as printed). Original source indexed by AMR-021-0005. https://arxiv.org/abs/1503.05295.