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Register a new userComplex polynomial Bohnenblust--Hille inequality with polynomial bounds
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Diana Serrano-Rodr\\'iguez M
Publication date
2019
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English
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The Bohnenblust-Hille inequality and its variants have found applications in several areas of Mathematics and related fields. The control of the constants for the variant for complex $m$-homogeneous polynomials is of special interest for applications in Harmonic Analysis and Number Theory. Up to now, the best known estimates for its constants are dominated by $kappaleft(1+varepsilonright) ^{m}$, where $varepsilon>0$ is arbitrary and $kappa>0$ depends on the choice of $varepsilon$. For the special cases in which the number of variables in each monomial is bounded by some fixed number $M$, it has been shown that the optimal constant is dominated by a constant depending solely on $M$. In this note, based on a deep result of Bayart, we prove an inequality for any subset of the indices, showing how summability of arbitrary restrictions on monomials can be related to the combinatorial dimension associated with them.
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In this paper we prove that the complex polynomial Bohnenblust-Hille constant for $2$-homogeneous polynomials in ${mathbb C}^2$ is exactly $sqrt[4]{frac{3}{2}}$. We also give the exact value of the real polynomial Bohnenblust-Hille constant for $2$-homogeneous polynomials in ${mathbb R}^2$. Finally, we provide lower estimates for the real polynomial Bohnenblust-Hille constant for polynomials in ${mathbb R}^2$ of higher degrees.
For the scalar field $mathbb{K}=mathbb{R}$ or $mathbb{C}$, the multilinear Bohnenblust--Hille inequality asserts that there exists a sequence of positive scalars $(C_{mathbb{K},m})_{m=1}^{infty}$ such that %[(sumlimits_{i_{1},...,i_{m}=1}^{N}|U(e_{i_{^{1}}}%,...,e_{i_{m}})|^{frac{2m}{m+1}})^{frac{m+1}{2m}}leq C_{mathbb{K},m}sup_{z_{1},...,z_{m}inmathbb{D}^{N}}|U(z_{1},...,z_{m})|] for all $m$-linear form $U:mathbb{K}^{N}times...timesmathbb{K}% ^{N}rightarrowmathbb{K}$ and every positive integer $N$, where $(e_{i})_{i=1}^{N}$ denotes the canonical basis of $mathbb{K}^{N}$ and $mathbb{D}^{N}$ represents the open unit polydisk in $mathbb{K}^{N}$. Since its proof in 1931, the estimates for $C_{mathbb{K},m}$ have been improved in various papers. In 2012 it was shown that there exist constants $(C_{mathbb{K},m})_{m=1}^{infty}$ with subexponential growth satisfying the Bohnenblust-Hille inequality. However, these constants were obtained via a complicated recursive formula. In this paper, among other results, we obtain a closed (non-recursive) formula for these constants with subexponential growth.
We show that a recent interpolative new proof of the Bohnenblust--Hille inequality, when suitably handled, recovers its best known constants. This seems to be unexpectedly surprising since the known interpolative approaches only provide constants having exponential growth. This preprint is no longer an independent submission, it is now contained in the preprint arXiv 1310.2834.
A number of sharp inequalities are proved for the space ${mathcal P}left(^2Dleft(frac{pi}{4}right)right)$ of 2-homogeneous polynomials on ${mathbb R}^2$ endowed with the supremum norm on the sector $Dleft(frac{pi}{4}right):=left{e^{itheta}:thetain left[0,frac{pi}{4}right]right}$. Among the main results we can find sharp Bernstein and Markov inequalities and the calculation of the polarization constant and the unconditional constant of the canonical basis of the space ${mathcal P}left(^2Dleft(frac{pi}{4}right)right)$.
We present new lower bounds that show that a polynomial number of passes are necessary for solving some fundamental graph problems in the streaming model of computation. For instance, we show that any streaming algorithm that finds a weighted minimum $s$-$t$ cut in an $n$-vertex undirected graph requires $n^{2-o(1)}$ space unless it makes $n^{Omega(1)}$ passes over the stream. To prove our lower bounds, we introduce and analyze a new four-player communication problem that we refer to as the hidden-pointer chasing problem. This is a problem in spirit of the standard pointer chasing problem with the key difference that the pointers in this problem are hidden to players and finding each one of them requires solving another communication problem, namely the set intersection problem. Our lower bounds for graph problems are then obtained by reductions from the hidden-pointer chasing problem. Our hidden-pointer chasing problem appears flexible enough to find other applications and is therefore interesting in its own right. To showcase this, we further present an interesting application of this problem beyond streaming algorithms. Using a reduction from hidden-pointer chasing, we prove that any algorithm for submodular function minimization needs to make $n^{2-o(1)}$ value queries to the function unless it has a polynomial degree of adaptivity.
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