3,266 research outputs found
Computing sparse multiples of polynomials
We consider the problem of finding a sparse multiple of a polynomial. Given f
in F[x] of degree d over a field F, and a desired sparsity t, our goal is to
determine if there exists a multiple h in F[x] of f such that h has at most t
non-zero terms, and if so, to find such an h. When F=Q and t is constant, we
give a polynomial-time algorithm in d and the size of coefficients in h. When F
is a finite field, we show that the problem is at least as hard as determining
the multiplicative order of elements in an extension field of F (a problem
thought to have complexity similar to that of factoring integers), and this
lower bound is tight when t=2.Comment: Extended abstract appears in Proc. ISAAC 2010, pp. 266-278, LNCS 650
Computing sparse multiples of polynomials
We consider the problem of finding a sparse multiple of a polynomial. Given
a polynomial f ∈ F[x] of degree d over a field F, and a desired sparsity
t = O(1), our goal is to determine if there exists a multiple h ∈ F[x] of f
such that h has at most t non-zero terms, and if so, to find such an h.
When F = Q, we give a polynomial-time algorithm in d and the size of
coefficients in h. For finding binomial multiples we prove a polynomial bound
on the degree of the least degree binomial multiple independent of coefficient
size.
When F is a finite field, we show that the problem is at least as hard as
determining the multiplicative order of elements in an extension field of F
(a problem thought to have complexity similar to that of factoring integers),
and this lower bound is tight when t = 2
A Size-Free CLT for Poisson Multinomials and its Applications
An -Poisson Multinomial Distribution (PMD) is the distribution of the
sum of independent random vectors supported on the set of standard basis vectors in . We show
that any -PMD is -close in total
variation distance to the (appropriately discretized) multi-dimensional
Gaussian with the same first two moments, removing the dependence on from
the Central Limit Theorem of Valiant and Valiant. Interestingly, our CLT is
obtained by bootstrapping the Valiant-Valiant CLT itself through the structural
characterization of PMDs shown in recent work by Daskalakis, Kamath, and
Tzamos. In turn, our stronger CLT can be leveraged to obtain an efficient PTAS
for approximate Nash equilibria in anonymous games, significantly improving the
state of the art, and matching qualitatively the running time dependence on
and of the best known algorithm for two-strategy anonymous
games. Our new CLT also enables the construction of covers for the set of
-PMDs, which are proper and whose size is shown to be essentially
optimal. Our cover construction combines our CLT with the Shapley-Folkman
theorem and recent sparsification results for Laplacian matrices by Batson,
Spielman, and Srivastava. Our cover size lower bound is based on an algebraic
geometric construction. Finally, leveraging the structural properties of the
Fourier spectrum of PMDs we show that these distributions can be learned from
samples in -time, removing
the quasi-polynomial dependence of the running time on from the
algorithm of Daskalakis, Kamath, and Tzamos.Comment: To appear in STOC 201
Solving Polynomial Systems via a Stabilized Representation of Quotient Algebras
We consider the problem of finding the isolated common roots of a set of
polynomial functions defining a zero-dimensional ideal I in a ring R of
polynomials over C. We propose a general algebraic framework to find the
solutions and to compute the structure of the quotient ring R/I from the null
space of a Macaulay-type matrix. The affine dense, affine sparse, homogeneous
and multi-homogeneous cases are treated. In the presented framework, the
concept of a border basis is generalized by relaxing the conditions on the set
of basis elements. This allows for algorithms to adapt the choice of basis in
order to enhance the numerical stability. We present such an algorithm and show
numerical results
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