Asymptotically fast polynomial matrix algorithms for multivariable systems

We present the asymptotically fastest known algorithms for some basic problems on univariate polynomial matrices: rank, nullspace, determinant, generic inverse, reduced form. We show that they essentially can be reduced to two computer algebra techniques, minimal basis computations and matrix fraction expansion/reconstruction, and to polynomial matrix multiplication. Such reductions eventually imply that all these problems can be solved in about the same amount of time as polynomial matrix multiplication

Algebraic approach to q-deformed supersymmetric variants of the Hubbard model with pair hoppings

We construct two quantum spin chains Hamiltonians with quantum sl(2|1) invariance. These spin chains define variants of the Hubbard model and describe electron models with pair hoppings. A cubic algebra that admits the Birman-Wenzl-Murakami algebra as a quotient allows exact solvability of the periodic chain. The two Hamiltonians, respectively built using the distinguished and the fermionic bases of U_q(sl(2|1)) differ only in the boundary terms. They are actually equivalent, but the equivalence is non local. Reflection equations are solved to get exact solvability on open chains with non trivial boundary conditions. Two families of diagonal solutions are found. The centre and the Scasimirs of the quantum enveloping algebra of sl(2|1) appear as tools for the construction of exactly solvable Hamiltonians.Comment: 22 pages, LaTeX2e, uses amsfonts; some references added and typos correcte

The Multivariate Resultant is NP-hard in any Characteristic

The multivariate resultant is a fundamental tool of computational algebraic geometry. It can in particular be used to decide whether a system of n homogeneous equations in n variables is satisfiable (the resultant is a polynomial in the system's coefficients which vanishes if and only if the system is satisfiable). In this paper we present several NP-hardness results for testing whether a multivariate resultant vanishes, or equivalently for deciding whether a square system of homogeneous equations is satisfiable. Our main result is that testing the resultant for zero is NP-hard under deterministic reductions in any characteristic, for systems of low-degree polynomials with coefficients in the ground field (rather than in an extension). We also observe that in characteristic zero, this problem is in the Arthur-Merlin class AM if the generalized Riemann hypothesis holds true. In positive characteristic, the best upper bound remains PSPACE.Comment: 13 page

Bi-criteria Pipeline Mappings for Parallel Image Processing

Mapping workflow applications onto parallel platforms is a challenging problem, even for simple application patterns such as pipeline graphs. Several antagonistic criteria should be optimized, such as throughput and latency (or a combination). Typical applications include digital image processing, where images are processed in steady-state mode. In this paper, we study the mapping of a particular image processing application, the JPEG encoding. Mapping pipelined JPEG encoding onto parallel platforms is useful for instance for encoding Motion JPEG images. As the bi-criteria mapping problem is NP-complete, we concentrate on the evaluation and performance of polynomial heuristics