4 research outputs found
Visibly irreducible polynomials over finite fields
H. Lenstra has pointed out that a cubic polynomial of the form
(x-a)(x-b)(x-c) + r(x-d)(x-e), where {a,b,c,d,e} is some permutation of
{0,1,2,3,4}, is irreducible modulo 5 because every possible linear factor
divides one summand but not the other. We classify polynomials over finite
fields that admit an irreducibility proof with this structure.Comment: 11 pages. To appear in the American Mathematical Monthl
Canonical rings of Q-divisors on P^1
The canonical ring of
a divisor D on a curve X is a natural object of study; when D is a Q-divisor,
it has connections to projective embeddings of stacky curves and rings of
modular forms. We study the generators and relations of S_D for the simplest
curve X = P^1. When D contains at most two points, we give a complete
description of S_D; for general D, we give bounds on the generators and
relations. We also show that the generators (for at most five points) and a
Groebner basis of relations between them (for at most four points) depend only
on the coefficients in the divisor D, not its points or the characteristic of
the ground field; we conjecture that the minimal system of relations varies in
a similar way. Although stated in terms of algebraic geometry, our results are
proved by translating to the combinatorics of lattice points in simplices and
cones.Comment: 19 pages, 3 figure
Large orbits on Markoff-type K3 surfaces over finite fields
We study the surface in , a tri-involutive K3 (TIK3) surface. We explain a
phenomenon noticed by Fuchs, Litman, Silverman, and Tran: over a finite field
of order mod , the points of do not form a single
large orbit under the group generated by the three involutions fixing
two variables and a few other obvious symmetries, but rather admit a partition
into two -invariant subsets of roughly equal size. The phenomenon is
traced to an explicit double cover of the surface.Comment: 4 pages. Accepted at IMR
Diophantine approximation on conics
Given a conic over , it is natural to ask what real
points on are most difficult to approximate by rational points of
low height. For the analogous problem on the real line (for which the least
approximable number is the golden ratio, by Hurwitz's theorem), the
approximabilities comprise the classically studied Lagrange and Markoff
spectra, but work by Cha-Kim and Cha-Chapman-Gelb-Weiss shows that the spectra
of conics can vary. We provide notions of approximability, Lagrange spectrum,
and Markoff spectrum valid for a general and prove that their
behavior is exhausted by the special family of conics , which has symmetry by the modular group and whose Markoff
spectrum was studied in a different guise by A. Schmidt and Vulakh. The proof
proceeds by using the Gross-Lucianovic bijection to relate a conic to a
quaternionic subring of and
classifying invariant lattices in its -dimensional representation.Comment: 13 pp., incl. 2 table