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Title: Waring rank of monomial Nikhil Balaji University of Ulm AG1 Mittagsseminar (own work) D1, D2, D3, D4, D5, RG1, SWS, MMCIWe use this to send out email in the morning. AG Audience English
Date: Thursday, 5 July 2018 16:00 30 Minutes Saarbrücken E1 4 024
 The Waring rank of a degree-d multivariate polynomial F(x_1,... x_n) is the least number $r$ such that, F(x_1,... x_n) = c_1\ell_1 + c_2\ell_2 + ... + c_r\ell_r where \ell_i = \alpha_{i1}x_1 + \alpha_{i2}x_2 + ... \alpha_{in}x_n. It follows from a simple application (due to Kayal) of the classical partial derivative method of Nisan and Wigderson that the multilinear monomial $\prod_{i=1}^n x_i$ has Waring rank at least $2^n/\sqrt{n}$ and a result of Fischer gives an upper bound of $2^{n-1}$ A result due to Ranestad and Schreyer (2011) proves that the Waring rank of the multilinear monomial is exactly $2^{n-1}$. A follow up work of Carlini, Catalisano and Geramita (2012) shows that the Waring rank of a general monomial $\prod_{i=1}^n x_i^{d_i}$ is exactly $\prod_{i=2}^n (d_i + 1)$. Both these proofs crucially make use of the notion of Apolarity, a notion that dates back to Sylvester. We provide an alternate proof of both these results and discuss a few natural related questions that remain unresolved. Part of ongoing work with Nutan Limaye, Sai Sandeep, Srikanth Srinivasan and Thomas Thierauf.
Name(s): Nitin Saurabh