Bounds for Homogeneous Polynomial Zeros

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

A polynomial $F\in\mathbb Z[x_1,\ldots,x_n]$ is a homogeneous form of degree $d$ if every monomial with nonzero coefficient has total degree $d$. A nontrivial integral zero is a vector $\mathbf x\in\mathbb Z^n\setminus\{\mathbf0\}$ with $F(\mathbf x)=0$. Such a zero exists exactly when a nonzero rational zero exists, by multiplying rational coordinates by a common denominator. For each odd $d\geq3$, define $\nu(d)$ as the least nonnegative integer $t$ such that every such form in every number $n>t$ of variables has a nontrivial integral zero; Birch's theorem ensures that $\nu(d)$ is finite.

Statement recorded in the supplied source.

For every odd integer $d\geq3$, obtain explicit upper bounds for $\nu(d)$ and improve their dependence on $d$; that is, find quantitatively sharp functions $B(d)$ for which \[ n>B(d)\ \Longrightarrow\ \left(\forall\text{ homogeneous }F\in\mathbb Z[x_1,\ldots,x_n] \text{ of degree }d\right) \left(\exists\mathbf x\in\mathbb Z^n\setminus\{\mathbf0\}\right) F(\mathbf x)=0. \] No nonsingularity, diagonal form, or coefficient-size restriction is imposed. Bounds are requested uniformly over all forms of the given odd degree. [S2]

Short English statement

Bound how many variables suffice to force a nonzero integer zero of every homogeneous polynomial of a given odd degree.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 1098 (GREEN-098). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Amichai Lampert, Andrew Snowden and Tamar Ziegler, Polynomial Bounds for Birch’s Theorem, arXiv:2512.00697v1 (2025), §1, definition of $N(d,s)$ and discussion of general degree bounds. For $K=\mathbb Q$, the recorded single-form threshold is $\nu(d)=N(d,1)-1$; the paper distinguishes fixed-degree dependence on the number of forms from dependence on degree. https://arxiv.org/abs/2512.00697.
• S3 Ben Green, 100 Open Problems, maintained author list, Problem 98 and its comments, accessed 7 October 2026. https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf.