The CarberyWright is a seminal result about the anticoncentration of polynomials of Gaussian random variables. See e.g. https://arxiv.org/pdf/1507.00829.pdf, Theorem 1.4, for the precise statement. However, I cannot find any reference where an explicit estimate on the constant B on the r.h.s. of the inequality is given. Knowing this constant is crucial for the application I have in mind. Are there known estimates on it? I should also say that the polynomial I have in mind is of the form $p(g_1,g_2,\ldots,g_k)=\langle g_1\otimes g_2 \ldots\otimes g_k,A g_1\otimes g_2 \ldots\otimes g_k\rangle$, where $A\in\mathbb{R}^{d^k\times d^k}$, $\textrm{tr}(A)=0$, and the $g_i\in \mathbb{R}^d$ are random vectors with i.i.d. gaussian entries. Maybe this special structure helps to obtain better anticoncentration estimates. Any help would be appreciated!
Have you looked at the original paper by A.Carbery and J.Wright ? Theorem 8 page 244 is the famous inequality with a sharp constant.

$\begingroup$ what do you mean by sharp constant? I do not see an explicit constant in that Theorem. You mean this article: pdfs.semanticscholar.org/e997/… ,right? It only says "There exists an absolute constant C". Do we know what that constant is? $\endgroup$ – user134977 May 26 '20 at 12:09

$\begingroup$ The way it is stated suggests that C is independent of the polynomial. The constant C is numerical. $\endgroup$ – user69642 May 26 '20 at 12:22

$\begingroup$ Oh, now I understand the issue. Sure, the constant C is numerical, but my application requires bounding this constant. Note that the statement of the inequality only becomes nontrivial for $\alpha\leq C^{2d}$, where $d$ is the degree of the polynomial. In the application I have in mind, the degree of the polynomial grows, so the inequality is nontrivial only for exponentially small $\alpha$, with the exponent depending on $C$. That is why I actually want to know $C$ in the inequality. $\endgroup$ – user134977 May 27 '20 at 13:33

$\begingroup$ Then, I guess you should be able to verify its order of magnitude numerically by testing different polynomials in dimension 1 for the Gaussian since it does not depend on the dimension, neither on the degree, neither on the polynomial neither on the measure... $\endgroup$ – user69642 May 27 '20 at 17:45

1$\begingroup$ Ok! I thought you were interested in a nontrivial lower bound. $\endgroup$ – user69642 May 29 '20 at 8:10