Let \(P_1, \ldots , P_N\) be complex polynomials of \(d\) complex variables, \(\delta_1, \ldots ,\delta_N\) be positive numbers, and \(B \subset \mathbb{C}^d\) be the ball of radius \(R\) (in the standard Euclidian distance) centered at zero. The central result of the paper says that, if \[ \sum_{k=1}^N \delta_k^2 \deg P_k \leqslant R^2, \] then there is a point \(u \in B\) which is, for every \(k\), at Euclidian distance at least \(\delta_k\) from the zero set of \(P_k\).
Defining a polynomial plank as a set of points whose distance to the zero set of a given polynomial does not exceed a given number, one reformulates this result in the spirit of covering by planks theorems, like in Section 3 of the survey [K. Ball, Convex geometry and functional analysis, in: Handbook of the geometry of Banach spaces. Volume 1. Amsterdam: Elsevier. 161–194 (2001; Zbl 1017.46004)].
The proof uses maximization of \(F(z)=\left| e^{-z^2/2}P_1^{\delta_1^2}(z)\cdots P_N^{\delta_N^2}(z)\right|\) on \(B\).
In the particular case of \(\deg P_k = 1\) for all \(k\), one gets an extension of the famous complex plank theorem due to K. M. Ball [Bull. Lond. Math. Soc. 33, No. 4, 433–442 (2001; Zbl 1030.46008)] to the case of planks that are not necessarily centrally symmetric.
Another result deals with real polynomials. Let \(P_1, \ldots, P_N \in \mathbb R [x_1, \dots , x_d]\) be polynomials with nonzero restrictions on the unit sphere \(S \subset \mathbb R^d\) and \(\delta_1, \ldots, \delta_N > 0\) be such that \[ \sum_{k=1}^N \delta_k \deg P_k \leqslant \frac{1}{e}. \] Then there exists a point \(p \in S\) such that, for every \(k = 1, \dots ,N\), the point \(p\) is at angular distance at least \(\delta_k\) from the intersection of the zero set of \(P_k\) with \(S\).
An open question, equivalent to Conjecture 1.8 from the previous paper of the same authors [Int. Math. Res. Not. 2023, No. 13, 11684–11700 (2023; Zbl 1526.46007)], is whether the constant \(\frac{1}{e}\) in the above result can be substituted by \(\frac{\pi}{2}\).