Bernstein inequality for multivariate polynomial - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T04:59:01Z http://mathoverflow.net/feeds/question/12298 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/12298/bernstein-inequality-for-multivariate-polynomial Bernstein inequality for multivariate polynomial Nurdin Takenov 2010-01-19T10:57:16Z 2010-01-19T16:37:20Z <p>Let $P$ be a polynomial in $k$ variables with complex coefficients, and $\deg P=n$. If $k=1$ then there is Bernstein's inequality:$||P'||\le n||P||$, where $||Q||=\max_{|z|=1} |Q(z)|$. So, are there similar results for $k\ge 2$? For example, what is the best $f(n,k)$, such that inequality $||\frac{\partial P}{\partial z_1}||+\cdots + ||\frac{\partial P}{\partial z_k}||\le f(n,k)||P||$, where $||Q||=\max_{|z_1|=\cdots=|z_k|=1}|Q(z_1,\cdots,z_k)|$, holds?</p> http://mathoverflow.net/questions/12298/bernstein-inequality-for-multivariate-polynomial/12319#12319 Answer by 002 for Bernstein inequality for multivariate polynomial 002 2010-01-19T16:37:20Z 2010-01-19T16:37:20Z <p>Tung, S. H. Bernstein's theorem for the polydisc. Proc. Amer. Math. Soc. 85 (1982), no. 1, 73--76. MR0647901 (83h:32017)</p> <p>(from MR review): Let $P(z)$ be a polynomial of degree $N$ in $z=(z_1,\cdots,z_m)$; suppose that $|P(z)|\leq 1$ for $z\in U^m$; then $\|DP(z)\|\leq N$ for $z\in U^m$ where $\|DP(z)\|^2=\sum_{i=1}^m|\partial P/\partial z_i|^2$. </p> <p>Here $U^m$ is the polydisc. Same author proved Bernstein-type inequality for the ball, Tung, S. H. Extension of Bernšteĭn's theorem. Proc. Amer. Math. Soc. 83 (1981), no. 1, 103--106. MR0619992 (82k:32013)</p>