21,905 research outputs found
Interpolation in Valiant's theory
We investigate the following question: if a polynomial can be evaluated at
rational points by a polynomial-time boolean algorithm, does it have a
polynomial-size arithmetic circuit? We argue that this question is certainly
difficult. Answering it negatively would indeed imply that the constant-free
versions of the algebraic complexity classes VP and VNP defined by Valiant are
different. Answering this question positively would imply a transfer theorem
from boolean to algebraic complexity. Our proof method relies on Lagrange
interpolation and on recent results connecting the (boolean) counting hierarchy
to algebraic complexity classes. As a byproduct we obtain two additional
results: (i) The constant-free, degree-unbounded version of Valiant's
hypothesis that VP and VNP differ implies the degree-bounded version. This
result was previously known to hold for fields of positive characteristic only.
(ii) If exponential sums of easy to compute polynomials can be computed
efficiently, then the same is true of exponential products. We point out an
application of this result to the P=NP problem in the Blum-Shub-Smale model of
computation over the field of complex numbers.Comment: 13 page
Shallow Circuits with High-Powered Inputs
A polynomial identity testing algorithm must determine whether an input
polynomial (given for instance by an arithmetic circuit) is identically equal
to 0. In this paper, we show that a deterministic black-box identity testing
algorithm for (high-degree) univariate polynomials would imply a lower bound on
the arithmetic complexity of the permanent. The lower bounds that are known to
follow from derandomization of (low-degree) multivariate identity testing are
weaker. To obtain our lower bound it would be sufficient to derandomize
identity testing for polynomials of a very specific norm: sums of products of
sparse polynomials with sparse coefficients. This observation leads to new
versions of the Shub-Smale tau-conjecture on integer roots of univariate
polynomials. In particular, we show that a lower bound for the permanent would
follow if one could give a good enough bound on the number of real roots of
sums of products of sparse polynomials (Descartes' rule of signs gives such a
bound for sparse polynomials and products thereof). In this third version of
our paper we show that the same lower bound would follow even if one could only
prove a slightly superpolynomial upper bound on the number of real roots. This
is a consequence of a new result on reduction to depth 4 for arithmetic
circuits which we establish in a companion paper. We also show that an even
weaker bound on the number of real roots would suffice to obtain a lower bound
on the size of depth 4 circuits computing the permanent.Comment: A few typos correcte
Automatic enumeration of regular objects
We describe a framework for systematic enumeration of families combinatorial
structures which possess a certain regularity. More precisely, we describe how
to obtain the differential equations satisfied by their generating series.
These differential equations are then used to determine the initial counting
sequence and for asymptotic analysis. The key tool is the scalar product for
symmetric functions and that this operation preserves D-finiteness.Comment: Corrected for readability; To appear in the Journal of Integer
Sequence
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