448 research outputs found
Multiplicative and Exponential Variations of Orthomorphisms of Cyclic Groups
An orthomorphism is a permutation of for which
is also a permutation on . Eberhard,
Manners, Mrazovi\'c, showed that the number of such orthomorphisms is
for odd and zero otherwise.
In this paper we prove two analogs of these results where is
replaced by (a "multiplicative orthomorphism") or with
(an "exponential orthomorphism"). Namely, we show that no
multiplicative orthomorphisms exist for but that exponential
orthomorphisms exist whenever is twice a prime such that is
squarefree. In the latter case we then estimate the number of exponential
orthomorphisms.Comment: 11 pages, 1 figur
Mod-2 dihedral Galois representations of prime conductor
For all odd primes N up to 500000, we compute the action of the Hecke operator T_2 on the space S_2(Gamma_0(N), Q) and determine whether or not the reduction mod 2 (with respect to a suitable basis) has 0 and/or 1 as eigenvalues. We then partially explain the results in terms of class field theory and modular mod-2 Galois representations. As a byproduct, we obtain some nonexistence results on elliptic curves and modular forms with certain mod-2 reductions, extending prior results of Setzer, Hadano, and Kida
Computation of Poincare-Betti series for monomial rings
The multigraded Poincare-Betti series P_R^k(x_1,...,x_n; t) of a monomial
ring k[x_1,...,x_n]/ on a finite number of monomial generators has the form
(1+tx_1)(1+tx_2)...(1+tx_n)/b_(R,k)(x_1,...,x_n; t), where
b_(R,k)(x_1,...,x_n;t) is a polynomial depending only on the monomial set M and
the characteristic of the field k. I present a computer program designed to
calculate the polynomial b_(R,k) for a given field characteristic and a given
set of monomial generators.Comment: 8 pages, prepared for the School and Workshop on Algebraic geometry
and statistics at Politecnico Torino in September 200
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