On the Size of Homogeneous and of Depth-Four Formulas with Low Individual Degree
Résumé
Let r ≥ 1 be an integer. Let us call a polynomial f (x 1 , x 2 ,. .. , x N) ∈ F[x] a multi-r-ic polynomial if the degree of f with respect to any variable is at most r. (This generalizes the notion of multilinear polynomials.) We investigate the arithmetic circuits in which the output is syntactically forced to be a multi-r-ic polynomial and refer to these as multi-r-ic circuits. We prove lower bounds for several subclasses of such circuits, including the following. 1. An N Ω(log N) lower bound against homogeneous multi-r-ic formulas (for an explicit multi-r-ic polynomial on N variables). 2. An (n/r 1.1) Ω √ d/r lower bound against depth-four multi-r-ic circuits computing the polynomial IMM n,d corresponding to the product of d matrices of size n × n each. √ N) lower bound against depth-four multi-r-ic circuits computing an explicit multi-r-ic polynomial on N variables.
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