4 research outputs found
Discovering the roots: Uniform closure results for algebraic classes under factoring
Newton iteration (NI) is an almost 350 years old recursive formula that
approximates a simple root of a polynomial quite rapidly. We generalize it to a
matrix recurrence (allRootsNI) that approximates all the roots simultaneously.
In this form, the process yields a better circuit complexity in the case when
the number of roots is small but the multiplicities are exponentially
large. Our method sets up a linear system in unknowns and iteratively
builds the roots as formal power series. For an algebraic circuit
of size we prove that each factor has size at most a
polynomial in: and the degree of the squarefree part of . Consequently,
if is a -hard polynomial then any nonzero multiple
is equally hard for arbitrary positive 's, assuming
that is at most .
It is an old open question whether the class of poly()-sized formulas
(resp. algebraic branching programs) is closed under factoring. We show that
given a polynomial of degree and formula (resp. ABP) size
we can find a similar size formula (resp. ABP) factor in
randomized poly()-time. Consequently, if determinant requires
size formula, then the same can be said about any of its
nonzero multiples.
As part of our proofs, we identify a new property of multivariate polynomial
factorization. We show that under a random linear transformation ,
completely factors via power series roots. Moreover, the
factorization adapts well to circuit complexity analysis. This with allRootsNI
are the techniques that help us make progress towards the old open problems,
supplementing the large body of classical results and concepts in algebraic
circuit factorization (eg. Zassenhaus, J.NT 1969, Kaltofen, STOC 1985-7 \&
Burgisser, FOCS 2001).Comment: 33 Pages, No figure
Stronger Lower Bounds and Randomness-Hardness Trade-Offs Using Associated Algebraic Complexity Classes
We associate to each Boolean language complexity class C the algebraic class a·C consisting of families of polynomials {fn} for which the evaluation problem over Z is in C. We prove the following lower bound and randomness-to-hardness results: 1. If polynomial identity testing (PIT) is in NSUBEXP then a·NEXP does not have poly size constant-free arithmetic circuits. 2. a·NEXP RP does not have poly size constant-free arithmetic circuits. 3. For every fixed k, a·MA does not have arithmetic circuits of size nk. Items 1 and 2 strengthen two results due to Kabanets and Impagliazzo [7]. The third item improves a lower bound due to Santhanam [11]. We consider the special case low-PIT of identity testing for (constant-free) arithmetic circuits with low formal degree, and give improved hardness-to-randomness trade-offs that apply to this case. Combining our results for both directions of the hardness-randomness connection, we demonstrate a case where derandomization of PIT and proving lower bounds are equivalent. Namely, we show that low-PIT ∈ i.o-NTIME[2no(1)]/no(1) if and only if there exists a family of multilinear polynomials in a·NE/lin that requires constant-free arithmetic circuits of super-polynomial size and formal degree