A note on separation of algebraic sets and the ?ojasiewicz exponent for polynomial mappings

In the paper a global separation problem for affine algebraic sets is considered. As application an upper bound for the distance of the graph of polynomial mapping to its zero set in the form of a ?ojasiewicz inequality is given.     

Gauss-Bruhat decomposition as an example of Thomas decomposition

Both the Gauss-Bruhat decomposition and the LU-decomposition of the general linear group over a field are examples of a Thomas decomposition of systems of polynomial equations and inequations into disjoint triangular systems, a recently rediscovered method of the nineteen-thirties, applied to the inequation det (A) ≠ 0 for an n × n-matrix of indeterminants. More specifically it is shown that the cells of the two decompositions can be described by determinantal equations and inequations yielding simple systems in the sense of Thomas of a rather special type, which are called split and allow counting solutions over any finite field. Received: 17 March 2008, Revised: 12 August 2008     

Polynomial images of R

Let R be a real closed field and n?2. We prove that: (1) for every finite subset F of Rⁿ, the semialgebraic set Rⁿ?F is a polynomial image of Rⁿ; and (2) for any independent linear forms l₁,…,l_r of Rⁿ, the semialgebraic set {l₁>0,…,l_r>0}⊂Rⁿ is a polynomial image of Rⁿ.     

Hilbert curves of polarized varieties

Let X be a normal Gorenstein complex projective variety. We introduce the Hilbert variety V_X associated to the Hilbert polynomial χ(x₁L₁+?+x_ρL_ρ), where L₁,…,L_ρ is a basis of , ρ being the Picard number of X, and x₁,…,x_ρ are complex variables. After studying general properties of V_X we specialize to the Hilbert curve of a polarized variety (X,L), namely the plane curve of degree dim(X) associated to χ(xK_X+yL). Special emphasis is given to the case of polarized threefolds.     

On open real polynomial maps

We study open polynomial maps from ⁿ to ^p. For n = p we give a complete characterization, and for p = 2, n ≥ 3 we obtain some partial information.     

On H. Weyl and J. Steiner Polynomials

The paper deals with root location problems for two classes of univariate polynomials both of geometric origin. The first class discussed, the class of Steiner polynomial, consists of polynomials, each associated with a compact convex set . A polynomial of this class describes the volume of the set V + tB ⁿ as a function of t, where t is a positive number and B ⁿ denotes the unit ball in . The second class, the class of Weyl polynomials, consists of polynomials, each associated with a Riemannian manifold , where is isometrically embedded with positive codimension in . A Weyl polynomial describes the volume of a tubular neighborhood of its associated as a function of the tube’s radius. These polynomials are calculated explicitly in a number of natural examples such as balls, cubes, squeezed cylinders. Furthermore, we examine how the above mentioned polynomials are related to one another and how they depend on the standard embedding of into for m > n. We find that in some cases the real part of any Steiner polynomial root will be negative. In certain other cases, a Steiner polynomial will have only real negative roots. In all of this cases, it can be shown that all of a Weyl polynomial’s roots are simple and, furthermore, that they lie on the imaginary axis. At the same time, in certain cases the above pattern does not hold.

Erasmus Darwin, the nephew of the great scientist Charles Darwin, believed that sometimes one should perform the most unusual experiments. They usually yield no results but when they do . . . . So once he played trumpet in front of tulips for the whole day. The experiment yielded no results.

Submitted: March 5, 2007., Revised: February 1, 2008., Accepted: February 2, 2008.     

Combinatorial bounds on Hilbert functions of fat points in projective space

We study Hilbert functions of certain non-reduced schemes A supported at finite sets of points in , in particular, fat point schemes. We give combinatorially defined upper and lower bounds for the Hilbert function of A using nothing more than the multiplicities of the points and information about which subsets of the points are linearly dependent. When N=2, we give these bounds explicitly and we give a sufficient criterion for the upper and lower bounds to be equal. When this criterion is satisfied, we give both a simple formula for the Hilbert function and combinatorially defined upper and lower bounds on the graded Betti numbers for the ideal I_A defining A, generalizing results of Geramita et al. (2006) [16]. We obtain the exact Hilbert functions and graded Betti numbers for many families of examples, interesting combinatorially, geometrically, and algebraically. Our method works in any characteristic.     

Sweeping algebraic curves for singular solutions

Many problems give rise to polynomial systems. These systems often have several parameters and we are interested to study how the solutions vary when we change the values for the parameters. Using predictor-corrector methods we track the solution paths. A point along a solution path is critical when the Jacobian matrix is rank deficient. The simplest case of quadratic turning points is well understood, but these methods no longer work for general types of singularities. In order not to miss any singular solutions along a path we propose to monitor the determinant of the Jacobian matrix. We examine the operation range of deflation and relate the effectiveness of deflation to the winding number. Computational experiments on systems coming from different application fields are presented.     

Impossibility of extending Pólya’s theorem to “forms” with arbitrary real exponents

Pólya proved that if a form (homogeneous polynomial) with real coefficients is positive on the nonnegative orthant (except at the origin), then it is the quotient of two real forms with no negative coefficients. While Pólya’s theorem extends, easily, from ordinary real forms to “generalized” real forms with arbitrary rational exponents, we show that it does not extend to generalized real forms with arbitrary real (possibly irrational) exponents.     

Asymptotic behaviour of families of real curves

We consider the family of fibres of a polynomial function f on a smooth noncompact algebraic real surface and we characterise the regular fibres of f which are atypical due to their asymptotic behaviour at infinity. We compare to the similar problem in the complex case. Received: 5 May 1998 / Revised version: 20 March 1999     

Asymptotic values of polynomial mappings of the real plane

Using algebraically constructible functions we give a generically effective method to detect asymptotic values of polynomial mappings with finite fibers defined on the real plane. By asymptotic variety we mean the set of points at which the polynomial mapping fails to be proper.     

Discrete behavior of Seshadri constants on surfaces

Working over , we show that, apart possibly from a unique limit point, the possible values of multi-point Seshadri constants for general points on smooth projective surfaces form a discrete set. In addition to its theoretical interest, this result is of practical value, which we demonstrate by giving significantly improved explicit lower bounds for Seshadri constants on and new results about ample divisors on blow ups of at general points.     

A note on families of hyperelliptic curves

We give a stack-theoretic proof for some results on families of hyperelliptic curves. Received: 5 February 2008     

Extensions of regular mappings and the ?ojasiewicz exponent at infinity

Let F:V→Cm be a regular mapping, where V⊂Cn is an algebraic set of positive dimension and m?n?2, and let L_∞(F) be the ?ojasiewicz exponent at infinity of F. We prove that F has a polynomial extension G:Cn→Cm such L_∞(G)=L_∞(F). Moreover, we give an estimate of the degree of the extension G. Additionally, we prove that if then for any β∈Q, β?L_∞(F), the mapping F has a polynomial extension G with L_∞(G)=β. We also give an estimate of the degree of this extension.     

A note on symbolic and ordinary powers of homogeneous ideals

In this note we are interested in the graded modulesM _k=I^(k)/I^k and , whereI is a saturated ideal in the homogeneous coordinate ringS=K[x₀,…,x_n] of ℙⁿ,I ^(k) is the symbolic power and is the saturation of the ordinary power. Very little is known about these modules, and we provide a bound on their diameters, we compute the Hilbert functions and we study some characteristic submodules in the special case ofn+1 general points in ℙⁿ.

---

Sunto In questa nota siamo interessati ai moduli graduatiM _k=I(k)/I^k e , doveI è un ideale saturato nell'anello delle coordinate omogeneeS:=K[x₀,…,x_n] di ℙⁿ,I ^(k) è la potenza simbolica e è la saturazione della potenza ordinaria. Poco è noto su questi moduli e qui viene fornito un limite superiore ai loro diametri. Ne calcoliamo inoltre le funzioni di Hilbert e studiamo alcuni sottomoduli caratteristici nel caso speciale din+1 punti in posizione generale, in ℙⁿ.

     

On the diameter and girth of a zero-divisor graph

For a commutative ring R with zero-divisors Z(R), the zero-divisor graph of R is Γ(R)=Z(R)−{0}, with distinct vertices x and y adjacent if and only if xy=0. In this paper, we characterize when either or . We then use these results to investigate the diameter and girth for the zero-divisor graphs of polynomial rings, power series rings, and idealizations.     

Minimizing setups in ordered sets of fixed width

A simply polynomial time algorithm is given for computing the setup number, or jump number, of an ordered set with fixed width. This arises as an interesting application of a polynomial time algorithm for solving a more general weighted problem in precedence constrained scheduling.     

Effective formulas for the ?ojasiewicz exponent at infinity

We give effective formulas for the ?ojasiewicz exponent at infinity of an arbitrary complex polynomial mapping.     

Cohen–Macaulayness and squarefree modules

In this paper we consider the category of squarefree modules over the polynomial ring and an exact duality functor, which is an extension of the Alexander dual of a simplicial complex. We give a relationship between the squarefree components of local cohomology groups of a squarefree module and the Tor groups of its dual. With this result it is shown that a squarefree module is sequentially Cohen–Macaulay if and only if the dual is componentwise linear. Received: 7 June 1999 / Revised version: 6 September 2000