On the Size of Homogeneous and of Depth-Four Formulas with Low Individual Degree

Neeraj Kayal (1) , Chandan Saha (2) , Sébastien Tavenas (1, 3)

1 Microsoft Research India [Bangalore]
2 IISc Bangalore - Indian Institute of Science [Bangalore]
3 LAMA - Laboratoire de Mathématiques

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.

Mots clés: Algebraic complexity; Arithmetic formulas; Arithmetic circuits; Partial derivatives

Domaines: Complexité [cs.CC]

Neeraj Kayal, Chandan Saha, Sébastien Tavenas. On the Size of Homogeneous and of Depth-Four Formulas with Low Individual Degree. Theory of Computing, 2018, 14 (1), pp.1 - 46. ⟨10.4086/toc.2018.v014a016⟩. ⟨hal-02447391⟩