Randomness Efficient Testing of Sparse Black Box Identities of Unbounded Degree over the Reals

Markus Bläser (1) , Christian Engels (1)

1 FR Informatik

We construct a hitting set generator for sparse multivariate polynomials over the reals. The seed length of our generator is $\Oh(\log^2 (mn/\epsilon))$ where $m$ is the number of monomials, $n$ is number of variables, and $1 - \epsilon$ is the hitting probability. The generator can be evaluated in time polynomial in $\log m$, $n$, and $\log 1/\epsilon$. This is the first hitting set generator whose seed length is independent of the degree of the polynomial. The seed length of the best generator so far by Klivans and Spielman \cite{KlivansSpielman01} depends logarithmically on the degree. From this, we get a randomized algorithm for testing sparse black box polynomial identities over the reals using $\Oh(\log^2 (mn/\epsilon))$ random bits with running time polynomial in $\log m$, $n$, and $\log \frac 1 \epsilon$. We also design a deterministic test with running time $\tilde \Oh(m^3 n^3)$. Here, the $\tilde \Oh$-notation suppresses polylogarithmic factors. The previously best deterministic test by Lipton and Vishnoi \cite{Lipton03} has a running time that depends polynomially on $\log \delta$, where $\delta$ is the degree of the black box polynomial.

Mots clés: Descartes' rule of signs; polynomial identity testing; sparse polynomials; black box testing

Domaines: Complexité [cs.CC]; Algorithme et structure de données [cs.DS]

Markus Bläser, Christian Engels. Randomness Efficient Testing of Sparse Black Box Identities of Unbounded Degree over the Reals. Symposium on Theoretical Aspects of Computer Science (STACS2011), Mar 2011, Dortmund, Germany. pp.555-566. ⟨hal-00573621⟩