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Dimension-free Markov--Bernstein-type inequality for ℝ2-valued polynomials

  • Egor Kosov
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Abstract

We prove a dimension-free Markov--Bernstein-type inequality for polynomial mappings (f1,f2):ℝn→ℝ2 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 ∇ f1 and ∇ f2 in terms of the Gram determinant of the d-th derivative tensors Ddf1 and Ddf2, 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.

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Cite

BibTeXRIS
BibTeX
@misc{kosov2026dimensionfree,
  title = {{Dimension-free Markov--Bernstein-type inequality for $\mathbb R^2$-valued polynomials}},
  author = {Egor Kosov},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.13836},
  eprint = {2610.13836},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.13836},
}
RIS
TY  - UNPB
TI  - Dimension-free Markov--Bernstein-type inequality for ℝ^2-valued polynomials
AU  - Egor Kosov
PY  - 2026
DA  - 2026/10/09
PB  - arXiv
DO  - 10.48550/arxiv.2610.13836
UR  - https://arxiv.org/abs/2610.13836
AB  - We prove a dimension-free Markov--Bernstein-type inequality for polynomial mappings (f_1,f_2):ℝ^n→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 ∇ f_1 and ∇ f_2 in terms of the Gram determinant of the d-th derivative tensors D^df_1 and D^df_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.
ER  -