Diagonalizations over polynomial time computable sets

AbstractA formal notion of diagonalization is developed which allows to enforce properties that are related to the class of polynomial time computable sets (the class of polynomial time computable functions respectively), like, e.g., p-immunity. It is shown that there are sets—called p-generic— which have all properties enforceable by such diagonalizations. We study the behaviour and the complexity of p-generic sets. In particular, we show that the existence of p-generic sets in NP is oracle dependent, even if we assume P ≠ NP

Asymptotic Hilbert Polynomial and limiting shapes

The main aim of this paper is to provide a method which allows finding limiting shapes of symbolic generic initial systems of higher-dimensional subvarieties of P^n. M. Mustata and S. Mayes established a connection between volumes of complements of limiting shapes and the asymptotic multiplicity for ideals of points. In the paper we prove a generalization of this fact to higher-dimensional sets

Special apolar subset: the case of star configurations

In this paper we consider a generic degree $d$ form $F$ in $n+1$ variables. In particular, we investigate the existence of star configurations apolar to $F$, that is the existence of apolar sets of points obtained by the $n$-wise intersection of $r$ general hyperplanes of $\mathbb{P}^n$. We present a complete answer for all values of $(d,r,n)$ except for $(d,d+1,2)$ when we present an algorithmic approach

Complex-temperature phase diagram of Potts and RSOS models

We study the phase diagram of Q-state Potts models, for Q=4 cos^2(PI/p) a Beraha number (p>2 integer), in the complex-temperature plane. The models are defined on L x N strips of the square or triangular lattice, with boundary conditions on the Potts spins that are periodic in the longitudinal (N) direction and free or fixed in the transverse (L) direction. The relevant partition functions can then be computed as sums over partition functions of an A\_{p-1} type RSOS model, thus making contact with the theory of quantum groups. We compute the accumulation sets, as N -> infinity, of partition function zeros for p=4,5,6,infinity and L=2,3,4 and study selected features for p>6 and/or L>4. This information enables us to formulate several conjectures about the thermodynamic limit, L -> infinity, of these accumulation sets. The resulting phase diagrams are quite different from those of the generic case (irrational p). For free transverse boundary conditions, the partition function zeros are found to be dense in large parts of the complex plane, even for the Ising model (p=4). We show how this feature is modified by taking fixed transverse boundary conditions.Comment: 60 pages, 16 figures, 2 table

The p-adic analytic space of pseudocharacters of a profinite group and pseudorepresentations over arbitrary rings

Let G be a profinite group which is topologically finitely generated, p a prime number and d an integer. We show that the functor from rigid analytic spaces over Q_p to sets, which associates to a rigid space Y the set of continuous d-dimensional pseudocharacters G -> O(Y), is representable by a quasi-Stein rigid analytic space X, and we study its general properties. Our main tool is a theory of "determinants" extending the one of pseudocharacters but which works over an arbitrary base ring; an independent aim of this paper is to expose the main facts of this theory. The moduli space X is constructed as the generic fiber of the moduli formal scheme of continuous formal determinants on G of dimension d. As an application to number theory, this provides a framework to study the generic fibers of pseudodeformation rings (e.g. of Galois representations), especially in the "residually reducible" case, and including when p <= d.Comment: 56 pages. v2 : final version, to appear in the Proceedings of the LMS Durham Symposium "Automorphic forms and Galois representations" (2011

Algebraic Bethe ansatz for the quantum group invariant open XXZ chain at roots of unity

For generic values of q, all the eigenvectors of the transfer matrix of the U_q sl(2)-invariant open spin-1/2 XXZ chain with finite length N can be constructed using the algebraic Bethe ansatz (ABA) formalism of Sklyanin. However, when q is a root of unity (q=exp(i pi/p) with integer p>1), the Bethe equations acquire continuous solutions, and the transfer matrix develops Jordan cells. Hence, there appear eigenvectors of two new types: eigenvectors corresponding to continuous solutions (exact complete p-strings), and generalized eigenvectors. We propose general ABA constructions for these two new types of eigenvectors. We present many explicit examples, and we construct complete sets of (generalized) eigenvectors for various values of p and N.Comment: 50pp, 2 figures, v2: few typos are fixed, Nucl. Phys. B (2016