6,152 research outputs found
A note on Agrawal conjecture
We prove that Lenstra proposition suggesting existence of many counterexamples to Agrawal conjecture is true in a more general case. At the same time we obtain a strictly ascending chain of subgroups of the group (Zp[X]/(Cr(X)))* and state the modified conjecture that the set {X-1, X+2} generate big enough subgroup of this group
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
Coefficient estimates for some classes of functions associated with -function theory
In this paper, for every , we obtain the Herglotz representation
theorem and discuss the Bieberbach type problem for the class of -convex
functions of order . In addition, we discuss the
Fekete-szeg\"o problem and the Hankel determinant problem for the class of
-starlike functions, leading to couple of conjectures for the class of
-starlike functions of order .Comment: 12 page
A generalization of starlike functions of order alpha
For every and we define a class of analytic
functions, the so-called -starlike functions of order , on the open
unit disk. We study this class of functions and explore some inclusion
properties with the well-known class of starlike functions of order .
The paper is also devoted to the discussion on the Herglotz representation
formula for analytic functions when is -starlike of
order . As an application we also discuss the Bieberbach conjecture
problem for the -starlike functions of order . Further application
includes the study of the order of -starlikeness of the well-known basic
hypergeometric functions introduced by Heine.Comment: 13 pages, 4 figures, submitted to a journa
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