17 research outputs found
Chebyshev Series Expansion of Inverse Polynomials
An inverse polynomial has a Chebyshev series expansion
1/\sum(j=0..k)b_j*T_j(x)=\sum'(n=0..oo) a_n*T_n(x) if the polynomial has no
roots in [-1,1]. If the inverse polynomial is decomposed into partial
fractions, the a_n are linear combinations of simple functions of the
polynomial roots. If the first k of the coefficients a_n are known, the others
become linear combinations of these with expansion coefficients derived
recursively from the b_j's. On a closely related theme, finding a polynomial
with minimum relative error towards a given f(x) is approximately equivalent to
finding the b_j in f(x)/sum_(j=0..k)b_j*T_j(x)=1+sum_(n=k+1..oo) a_n*T_n(x),
and may be handled with a Newton method providing the Chebyshev expansion of
f(x) is known.Comment: LaTeX2e, 24 pages, 1 PostScript figure. More references. Corrected
typos in (1.1), (3.4), (4.2), (A.5), (E.8) and (E.11
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