Dimension-free Markov--Bernstein-type inequality for ${\mathbb{R}}^2$-valued polynomials

We prove a dimension-free Markov–Bernstein-type inequality for polynomial mappings $(f_1,f_2)\text{\hspace{0.17em}⁣}:{\mathbb{R}}^n\to {\mathbb{R}}^2$ of degree at most $d$. For product probability measures whose one-dimensional marginals have densities bounded by $1$, we obtain a lower bound for the integral of the Gram determinant of $\mathrm{\nabla }{f}_1$ and $\mathrm{\nabla }{f}_2$ in terms of the Gram determinant of the $d$-th derivative tensors $D^d{f}_1$ and $D^d{f}_2$, with a constant depending only on $d$. For pairs in a fixed Wiener chaos, the resulting Gaussian inequality answers a question of Nourdin, Nualart, and Poly concerning a lower bound for the expectation of the Malliavin determinant in terms of the determinant of the covariance matrix.