Dvoretzky type theorems for multivariate polynomials and sections of convex bodies

TL;DR

This paper proves a Dvoretzky-type theorem for multivariate polynomials, confirming a conjecture for certain cases and improving bounds, with implications for convex body sections and projections.

Contribution

It proves the Gromov--Milman conjecture for homogeneous polynomials and advances bounds for the Birch theorem in odd degrees and complex polynomials.

Findings

Proved the Gromov--Milman conjecture for homogeneous polynomials.

Improved bounds on the number n(d,k) for odd degrees and complex polynomials.

Derived corollaries for John ellipsoids of convex body sections.

Abstract

In this paper we prove the Gromov--Milman conjecture (the Dvoretzky type theorem) for homogeneous polynomials on $\mathbb R^n$, and improve bounds on the number $n(d,k)$ in the analogous conjecture for odd degrees $d$ (this case is known as the Birch theorem) and complex polynomials. We also consider a stronger conjecture on the homogeneous polynomial fields in the canonical bundle over real and complex Grassmannians. This conjecture is much stronger and false in general, but it is proved in the cases of $d=2$ (for $k$'s of certain type), odd $d$, and the complex Grassmannian (for odd and even $d$ and any $k$). Corollaries for the John ellipsoid of projections or sections of a convex body are deduced from the case $d=2$ of the polynomial field conjecture.