2 research outputs found
Log-concavity and lower bounds for arithmetic circuits
One question that we investigate in this paper is, how can we build
log-concave polynomials using sparse polynomials as building blocks? More
precisely, let be a
polynomial satisfying the log-concavity condition a\_i^2 \textgreater{} \tau
a\_{i-1}a\_{i+1} for every where \tau
\textgreater{} 0. Whenever can be written under the form where the polynomials have at most
monomials, it is clear that . Assuming that the
have only non-negative coefficients, we improve this degree bound to if \tau \textgreater{} 1,
and to if .
This investigation has a complexity-theoretic motivation: we show that a
suitable strengthening of the above results would imply a separation of the
algebraic complexity classes VP and VNP. As they currently stand, these results
are strong enough to provide a new example of a family of polynomials in VNP
which cannot be computed by monotone arithmetic circuits of polynomial size