On Geometric Complexity Theory: Multiplicity Obstructions Are Stronger Than Occurrence Obstructions

Julian Dörfler, Christian Ikenmeyer, Greta Panova
2020 SIAM Journal on applied algebra and geometry  
Geometric Complexity Theory as initiated by Mulmuley and Sohoni in two papers (SIAM J Comput 2001 , 2008 aims to separate algebraic complexity classes via representation theoretic multiplicities in coordinate rings of specific group varieties. We provide the first toy setting in which a separation can be achieved for a family of polynomials via these multiplicities. Mulmuley and Sohoni's papers also conjecture that the vanishing behavior of multiplicities would be sufficient to separate
more » ... ty classes (so-called occurrence obstructions). The existence of such strong occurrence obstructions has been recently disproven in 2016 in two successive papers, Ikenmeyer-Panova (Adv. Math.) and Bürgisser-Ikenmeyer-Panova (J. AMS). This raises the question whether separating group varieties via representation theoretic multiplicities is stronger than separating them via occurrences. We provide first finite settings where a separation via multiplicities can be achieved, while the separation via occurrences is provably impossible. These settings are surprisingly simple and natural: We study the variety of products of homogeneous linear forms (the so-called Chow variety) and the variety of polynomials of bounded border Waring rank (i.e. a higher secant variety of the Veronese variety). As a side result we prove a slight generalization of Hermite's reciprocity theorem, which proves Foulkes' conjecture for a new infinite family of cases. ACM Subject Classification Theory of computation → Algebraic complexity theory
doi:10.1137/19m1287638 fatcat:2ybwbv6s2rbmdfbbghz5jtphia