The Warp machine is a systolic array computer of linearly connected cells, each of which is a programmable processor capable of performing 10 million floating-point operations per second (10 MFLOPS). A typical Warp array includes 10 cells, thus having a peak computation rate of 100 MFLOPS. The Warp array can be extended to include more cells to accommodate applications capable of using the increased computational bandwidth. Warp is integrated as an attached processor into a UN host system. Programs for Warp are written in a high-level language supported by an optimizing compiler.

Let F be a set of polynomials in the variables x = x1,..., xn with coefficients in R[a], where R is a UFD and a = a1,..., am a set of parameters. In this paper we present a new algorithm for discussing Gröbner bases with parameters. The algorithm obtains all the cases over the parameters leading to different reduced Gröbner basis, when the parameters in F are substituted in an extension field K of R. This new algorithm improves Weispfenning’s comprehensive Gröbner basis CGB algorithm, obtaining a reduced complete set of compatible and disjoint cases. A final improvement determines the minimal singular variety outside of which the Gröbner basis of the generic case specializes properly. These constructive methods provide a very satisfactory discussion and rich geometrical interpretation in the applications. c ○ 2002 Academic Press 1.

The algebraic technique of Gröbner bases is applied to study triangulations of the second hypersimplex \Delta(2; n). We present a quadratic Gröbner basis for the associated toric ideal I(Kn ). The simplices in the resulting triangulation of \Delta(2; n) have unit volume, and they are indexed by subgraphs which are linear thrackles [28] with respect to a circular embedding of Kn . For n 6 the number of distinct initial ideals of I(Kn ) exceeds the number of regular triangulations of \Delta(2; n); more precisely, the secondary polytope of \Delta(2; n) equals the state polytope of I(Kn ) for n 5 but not for n 6. We also construct a non-regular triangulation of \Delta(2; n) for n 9. We determine an explicit universal Gröbner basis of I(Kn ) for n 8. Potential applications in combinatorial optimization and random generation of graphs are indicated.

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Systems of polynomial equations over the complex or real numbers can be used to model combinatorial problems. In this way, a combinatorial problem is feasible (e.g. a graph is 3-colorable, hamiltonian, etc.) if and only if a related system of polynomial equations has a solution. In the first part of this paper, we construct new polynomial encodings for the problems of finding in a graph its longest cycle, the largest planar subgraph, the edge-chromatic number, or the largest k-colorable subgraph. For an infeasible polynomial system, the (complex) Hilbert Nullstellensatz gives a certificate that the associated combinatorial problem is infeasible. Thus, unless P = NP, there must exist an infinite sequence of infeasible instances of each hard combinatorial problem for which the minimum degree of a Hilbert Nullstellensatz certificate of the associated polynomial system grows. We show that the minimum-degree of a Nullstellensatz certificate for the non-existence of a stable set of size greater than the stability number of the graph is the stability number of the graph. Moreover, such a certificate contains at least one term per stable set of G. In contrast, for non-3-colorability, we found only graphs with Nullstellensatz certificates of degree four.

This paper addresses the problems of detecting Hopf bifurcations in systems of ordinary differential equations and following curves of Hopf points in two parameter families of vector fields. The established approach to this problem relies upon augmenting the equilibrium condition so that a Hopf bifurcation occurs at an isolated, regular point of the extended system. We propose two new methods of this type, based on classical algebraic results regarding the roots of polynomial equations and properties of Kronecker products for matrices. In addition to their utility as augmented systems for use with standard Newton-type continuation methods, they are also particularly well-adapted for solution by computer algebra techniques for vector fields of small or moderate dimension.

Systems of polynomial equations over an algebraically-closed field K can be used to concisely model many combinatorial problems. In this way, a combinatorial problem is feasible (e.g., a graph is 3-colorable, hamiltonian, etc.) if and only if a related system of polynomial equations has a solution over K. In this paper, we investigate an algorithm aimed at proving combinatorial infeasibility based on the observed low degree of Hilbert’s Nullstellensatz certificates for polynomial systems arising in combinatorics and on large-scale linear-algebra computations over K. We report on experiments based on the problem of proving the non-3-colorability of graphs. We successfully solved graph problem instances having thousands of nodes and tens of thousands of edges.

this paper. Particularly, it is shown how the sparse resultant matrices can be constructed directly and easily from the Dixon formulation, without having to explicitly construct the support of a polynomial system. An algorithm for constructing sparse matrices from the polynomial system is given. Its complexity is analyzed. This algorithm is empirically and theoretically compared with the subdivision and incremental methods for constructing sparse resultant matrices proposed in [EC95, CE96]. Another important issue in elimination theory is that of extraneous factors in the projection operators computed from such sparse matrices. For a suite of examples, extraneous factors arising from the sparse matrices constructed using the proposed algorithm are compared with the extraneous factors arising from the sparse resultant matrices based on subdivision and incremental algorithms. In section 3.2, a structural relationship between sparse matrices and Dixon matrices arising from the Dixon formulation is explored. Such an analysis, it is believed, will reveal how sparse matrices so constructed might be useful in determining extraneous factors arising in projection operators computed from the generalized Dixon resultant formulation as discussed in [KSY94, Sax97]. 2 Background This section introduces some notation and denitions. A reader can nd a general introduction in [CLO96] and [CLO98]. Let C [x] = C [x 1 ; : : : ; x d ], and denote by x = x 1 1 x d d . Let P = ff 1 ; f 2 ; : : : ; f k g where f i = P c x 2 C [x]. In general, this polynomial system P of k equations can be expressed in matrix notation as P = 0 B B B @ c 1;1 c 1;2 c 1;n c 2;1 c 2;2 c 2;n . . . . . . . . . . .

This paper gives an algorithm for computing invariant rings of reductive groups in arbitrary characteristic. Previously, only algorithms for linearly reductive groups and for finite groups have been known. The key step is to find a separating set of invariants.

Systems of polynomial equations over an algebraically-closed field K can be used to easily model combinatorial problems. In this way, a combinatorial problem is feasible (e.g., a graph is 3colorable, hamiltonian, etc.) if and only if a related system of polynomial equations has a solution over K. In this paper we investigate an algorithm aimed at proving combinatorial infeasibility based on the low degree of Hilbert’s Nullstellensatz certificates for polynomial systems arising in combinatorics and large-scale linear algebra computations over K. We report on experiments based on the problem of proving the non-3-colorability of graphs. We successfully solved graph problem instances having thousands of nodes and tens of thousands of edges. 1