16,382 research outputs found
On the graph of a function over a prime field whose small powers have bounded degree
Let be a function from a finite field with a prime number of elements, to . In this article we consider those functions for which there is a positive integer with the property that , when considered as an element of , has degree at most , for all . We prove that every line is incident with at most points of the graph of , or at least points, where is a positive integer satisfying if is even and if is odd. With the additional hypothesis that there are lines that are incident with at least points of the graph of , we prove that the graph of is contained in these lines. We conjecture that the graph of is contained in an algebraic curve of degree and prove the conjecture for and . These results apply to functions that determine less than directions. In particular, the proof of the conjecture for and gives new proofs of the result of Lov\'asz and Schrijver \cite{LS1981} and the result in \cite{Gacs2003} respectively, which classify all functions which determine at most directions
Computing the endomorphism ring of an ordinary elliptic curve over a finite field
We present two algorithms to compute the endomorphism ring of an ordinary
elliptic curve E defined over a finite field F_q. Under suitable heuristic
assumptions, both have subexponential complexity. We bound the complexity of
the first algorithm in terms of log q, while our bound for the second algorithm
depends primarily on log |D_E|, where D_E is the discriminant of the order
isomorphic to End(E). As a byproduct, our method yields a short certificate
that may be used to verify that the endomorphism ring is as claimed.Comment: 16 pages (minor edits
The Galois Complexity of Graph Drawing: Why Numerical Solutions are Ubiquitous for Force-Directed, Spectral, and Circle Packing Drawings
Many well-known graph drawing techniques, including force directed drawings,
spectral graph layouts, multidimensional scaling, and circle packings, have
algebraic formulations. However, practical methods for producing such drawings
ubiquitously use iterative numerical approximations rather than constructing
and then solving algebraic expressions representing their exact solutions. To
explain this phenomenon, we use Galois theory to show that many variants of
these problems have solutions that cannot be expressed by nested radicals or
nested roots of low-degree polynomials. Hence, such solutions cannot be
computed exactly even in extended computational models that include such
operations.Comment: Graph Drawing 201
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