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In this paper we study and relate several invariants connected to the solving degree of a polynomial system. This provides a rigorous framework for estimating the complexity of solving a system of polynomial equations via Gröbner bases methods. Our main results include a connection between the solving degree and the last fall degree and one between the degree of regularity and the Castelnuovo–Mumford regularity. [ABSTRACT FROM AUTHOR]

The isotropic Grassmannian parametrizes isotropic subspaces of a vector space equipped with a quadratic form. In this paper, we show that any maximal isotropic Grassmannian in its Plücker embedding can be defined by pulling back the equations of G r iso (3 , 7) or G r iso (4 , 8). [ABSTRACT FROM AUTHOR]

The main object of study in this paper is the well-known Somos-4 recurrence. We prove a theorem that any sequence generated by this equation also satisfies Gale-Robinson one. The corresponding identity is written in terms of its companion elliptic sequence. An example of such relationship is provided by the second-order linear sequence which, as we prove using Wajda's identity, satisfies the Somos-4 recurrence with suitable coefficients. Also, we construct a class of solutions to Volterra lattice equation closely related to the second-order linear sequence. [ABSTRACT FROM AUTHOR]

The horospherical p -Minkowski problem in hyperbolic space H n + 1 was proposed by Li-Xu in [14]. In particular, when n = 1 , p = − 1 , it is called the Christoffel problem in the hyperbolic plane. The corresponding equation is given by φ (φ θ θ − φ θ 2 2 φ + φ − φ − 1 2) = f (θ) , θ ∈ [ 0 , 2 π). In this paper, we obtain two existence results of solutions to the above equation. [ABSTRACT FROM AUTHOR]

In this paper, we study the convergence properties of a Newton-type method for solving generalized equations under a majorant condition. To this end, we use a contraction mapping principle. More precisely, we present semi-local convergence analysis of the method for generalized equations involving a set-valued map, the inverse of which satisfying the Aubin property. Our analysis enables us to obtain convergence results under Lipschitz, Smale and Nesterov-Nemirovski's self-concordant conditions. [ABSTRACT FROM AUTHOR]

We improve the algorithms of Lauder-Wan [11] and Harvey [8] to compute the zeta function of a system of m polynomial equations in n variables, over the q element finite field F q , for large m. The dependence on m in the original algorithms was exponential in m. Our main result is a reduction of the dependence on m from exponential to polynomial. As an application, we speed up a doubly exponential algorithm from a recent software verification paper [3] (on universal equivalence of programs over finite fields) to singly exponential time. One key new ingredient is an effective, finite field version of the classical Kronecker theorem which (set-theoretically) reduces the number of defining equations for a polynomial system over F q when q is suitably large. [ABSTRACT FROM AUTHOR]