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TR13-011 | 10th January 2013 09:42

Multilinear Complexity is Equivalent to Optimal Tester Size


Authors: Nader Bshouty
Publication: 13th January 2013 09:51
Downloads: 2538


In this paper we first show that Tester for an $F$-algebra $A$
and multilinear forms (see Testers and their Applications ECCC 2012) is equivalent to multilinear
algorithm for the product of elements in $A$
(see Algebraic
complexity theory. vol. 315, Springer-Verlag). Our
result is constructive in deterministic polynomial time. We show
that given a tester of size $\nu$ for an $F$-algebra $A$
and multilinear forms of degree $d$ one can in deterministic
polynomial time construct a multilinear algorithm for the
multiplication of $d$ elements of the algebra of multilinear
complexity $\nu$ and vise versa.

This with the constructions in above paper give the first polynomial
time construction of a bilinear algorithm with linear bilinear
complexity for the multiplication of two elements in any extension
finite field.

We then study the problem of simulating a substitution of an
assignment from an $F$-algebra $A$ in a degree $d$
multivariate polynomials with substitution of assignments from the
ground field $F$. We give a complete classification of all
algebras for which this can be done and show that this problem is
equivalent to constructing symmetric multilinear
algorithms for the product of $d$ elements in $A$.

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