Set Of Homogeneous Polynomials Research Articles

Computation of Macaulay constants and degree bounds for Gröbner bases

In this paper, following the approach by Dubé (1990) and by applying the Hilbert series method, we provide an efficient algorithm to compute the Macaulay constants of the quotient ring of a monomial ideal without computing any exact cone decomposition for the quotient ring. Then, based on this construction and the method proposed by Mayr and Ritscher (2013), a new upper bound for the maximum degree of the elements of any reduced Gröbner basis of an ideal generated by a set of homogeneous polynomials is given. The new bound depends on the Krull dimension and the maximum degree of the generating set of the ideal. Finally, we show that the presented upper bound is sharper than the bounds proposed by Dubé (1990) and Mayr and Ritscher (2013).

Degree Upper Bounds for Involutive Bases

The aim of this paper is to investigate upper bounds for the maximum degree of the elements of any minimal Janet basis of an ideal generated by a set of homogeneous polynomials. The presented bounds depend on the number of variables and the maximum degree of the generating set of the ideal. For this purpose, by giving a deeper analysis of the method due to Dube (SIAM J Comput 19:750–773, 1990), we improve (and correct) his bound on the degrees of the elements of a reduced Grobner basis. By giving a simple proof, it is shown that this new bound is valid for Pommaret bases, as well. Furthermore, based on Dube’s method, and by introducing two new notions of genericity, so-called J-stable position and prime position, we show that Dube’s (new) bound holds also for the maximum degree of polynomials in any minimal Janet basis of a homogeneous ideal in any of these positions. Finally, we study the introduced generic positions by proposing deterministic algorithms to transform any given homogeneous ideal into these positions.

A new algorithm for computing SAGBI bases up to an arbitrary degree

We present a new algorithm for computing a SAGBI basis up to an arbitrary degree for a subalgebra generated by a set of homogeneous polynomials. Our idea is based on linear algebra methods which cause a low level of complexity and computational cost. We then use it to solve the membership problem in subalgebras.

Cartan’s conjecture for moving hypersurfaces

Let f be a holomorphic curve in $${\mathbb {P}}^n({{\mathbb {C}}})$$ and let $$\mathcal {D}=\{D_1,\ldots ,D_q\}$$ be a family of moving hypersurfaces defined by a set of homogeneous polynomials $$\mathcal {Q}=\{Q_1,\ldots ,Q_q\}$$ . For $$j=1,\ldots ,q$$ , denote by $$Q_j=\sum \nolimits _{i_0+\cdots +i_n=d_j}a_{j,I}(z)x_0^{i_0}\cdots x_n^{i_n}$$ , where $$I=(i_0,\ldots ,i_n)\in {\mathbb {Z}}_{\ge 0}^{n+1}$$ and $$a_{j,I}(z)$$ are entire functions on $${{\mathbb {C}}}$$ without common zeros. Let $$\mathcal {K}_{\mathcal {Q}}$$ be the smallest subfield of meromorphic function field $$\mathcal {M}$$ which contains $${{\mathbb {C}}}$$ and all $$\frac{a_{j,I'}(z)}{a_{j,I''}(z)}$$ with $$a_{j,I''}(z)\not \equiv 0$$ , $$1\le j\le q$$ . In previous known second main theorems for f and $$\mathcal {D}$$ , f is usually assumed to be algebraically nondegenerate over $$\mathcal {K}_{\mathcal {Q}}$$ . In this paper, we prove a second main theorem in which f is only assumed to be nonconstant. This result can be regarded as a generalization of Cartan’s conjecture for moving hypersurfaces.

Products of Admissible Monomials in the Polynomial Algebra as a Module over the Steenrod Algebra

Let ${\P}(n) ={\F}[x_1,\ldots,x_n]$ be the polynomial algebra in $n$ variables $x_i$, of degree one, over the field $\F$ of two elements. The mod-2 Steenrod algebra $\A$ acts on ${\P }(n)$ according to well known rules. A major problem in algebraic topology is that of determining $\A^+{\P}(n)$, the image of the action of the positively graded part of $\A$. We are interested in the related problem of determining a basis for the quotient vector space ${\Q}(n) = {\P}(n)/\A^{+}\P(n)$. Both ${\P }(n) =\bigoplus_{d \geq 0} {\P}^{d}(n)$ and ${\Q}(n)$ are graded, where ${\P}^{d}(n)$ denotes the set of homogeneous polynomials of degree $d$. ${\Q}(n)$ has been explicitly calculated for $n=1,2,3,4$ but problems remain for $n \geq 5.$ In this note we show that if $u = x_{1}^{m_1} \cdots x_{k}^{m_{k}} \in {\P}^{d}(k)$ and $v = x_{1}^{e_1} \cdots x_{r}^{e_{r}} \in {\P}^{d'}(r)$ are an admissible monomials, (that is, $u$ and $v$ meet a criterion to be in a certain basis for ${\Q}(k)$ and ${\Q}(r)$ respectively), then for each permutation $\sigma \in S_{k+r}$ for which $\sigma(i)<\sigma(j),$ $i<j\leq k$ and $\sigma(s)<\sigma(t),$ $k<s<t\leq k+r,$ the monomial $x_{\sigma(1)}^{m_1} \cdots x_{\sigma(k)}^{m_{k}} x_{\sigma(k+1)}^{e_1} \cdots x_{\sigma(k+r)}^{e_r} \in {\P}^{d+d'}(k+r)$ is admissible. As an application we consider a few cases when $n=5.$

Recovering Exact Results from Inexact Numerical Data in Algebraic Geometry

Let be a set of homogeneous polynomials. Let Z denote the complex projective algebraic set determined by the zero locus of . Numerical-continuation-based methods can be used to produce arbitrary-precision numerical approximations of generic points on each irreducible component of Z. Consider the ideal and the prime decomposition over . This article illustrates how lattice-reduction algorithms may take as input numerically approximated generic points on Z and effectively extract exact elements for each Pi . A collection of examples serves to illustrate the approach and indicate some of the application areas for which this technique is valuable.

Dimension Result for the Polynomial Algebra as a Module over the Steenrod Algebra

For let be the polynomial algebra in variables of degree one, over the field of two elements. The mod-2 Steenrod algebra acts on according to well-known rules. Let denote the image of the action of the positively graded part of A major problem is that of determining a basis for the quotient vector space Both and are graded where denotes the set of homogeneous polynomials of degree A spike of degree is a monomial of the form where for each In this paper we show that if and can be expressed in the form with then where is the number of spikes of degree

Invariant norm quantifying nonlinear content of Hamiltonian systems

Given a Hamiltonian system, one can represent it using a symplectic map. This symplectic map is specified by a set of homogeneous polynomials which are uniquely determined by the Hamiltonian. In this paper, we construct an invariant norm in the space of homogeneous polynomials of a given degree. This norm is a function of parameters characterizing the original Hamiltonian system. Such a norm has several potential applications.

A Weierstrass-type theorem for homogeneous polynomials

By the celebrated Weierstrass Theorem the set of algebraic polynomials is dense in the space of continuous functions on a compact set in $\mathbb {R}^d$. In this paper we study the following question: does the density hold if we approximate only by homogeneous polynomials? Since the set of homogeneous polynomials is nonlinear, this leads to a nontrivial problem. It is easy to see that: 1) density may hold only on star-like 0-symmetric surfaces; 2) at least 2 homogeneous polynomials are needed for approximation. The most interesting special case of a star-like surface is a convex surface. It has been conjectured by the second author that functions continuous on 0-symmetric convex surfaces in $\mathbb {R}^d$ can be approximated by sums of 2 homogeneous polynomials. This conjecture has not yet been resolved, but we make substantial progress towards its positive settlement. In particular, it is shown in the present paper that the above conjecture holds for 1) $d=2$; 2) convex surfaces in $\mathbb {R}^d$ with $C^{1+\epsilon }$ boundary.

Generalized principal component analysis (GPCA)

This paper presents an algebro-geometric solution to the problem of segmenting an unknown number of subspaces of unknown and varying dimensions from sample data points. We represent the subspaces with a set of homogeneous polynomials whose degree is the number of subspaces and whose derivatives at a data point give normal vectors to the subspace passing through the point. When the number of subspaces is known, we show that these polynomials can be estimated linearly from data; hence, subspace segmentation is reduced to classifying one point per subspace. We select these points optimally from the data set by minimizing certain distance function, thus dealing automatically with moderate noise in the data. A basis for the complement of each subspace is then recovered by applying standard PCA to the collection of derivatives (normal vectors). Extensions of GPCA that deal with data in a high-dimensional space and with an unknown number of subspaces are also presented. Our experiments on low-dimensional data show that GPCA outperforms existing algebraic algorithms based on polynomial factorization and provides a good initialization to iterative techniques such as K-subspaces and Expectation Maximization. We also present applications of GPCA to computer vision problems such as face clustering, temporal video segmentation, and 3D motion segmentation from point correspondences in multiple affine views.

Polynomials and multilinear mappings in topological modules

Criteria for the equicontinuity of sets of multilinear mappings between topological modules are studied, as well as topological modules of continuous multilinear mappings. As a consequence, criteria for the equicontinuity of sets of homogeneous polynomials between topological modules are also studied, as well as topological modules of continuous homogeneous polynomials.

A sequence of complexes generated by a finite set of homogeneous polynomials

The usefulness of the Koszul complex in handling in an algebraic setting the two geometric notions of multiplicity and depth first became apparent with the work of Auslander and Buchsbaum [1] following a suggestion of Serre. Regarding the generators a1, …, an of the complex as a 1×n matrix first Eagon and Northcott [4] extended this work to a complex associated with an m×n matrix, then shortly afterwards a different extension was given by Buchsbaum and Rim [2, 3]. These two complexes are two of an infinite family [6] some of which inherit the depth sensitive property of the Koszul complex and all of which under a certain finiteness condition provide the same multiplicity as Euler–Poincaré characteristic [7].These two properties prove useful in geometric applications, see for example Lago and Rodicio [7] for depth sensitivity and Kirby [8] for the characteristic as multiplicity of intersection. During the course of these developments it became clear that it was most appropriate to regard the complexes as being generated by linear forms. From this point of view it is natural to ask if the linearity of the forms is necessary. In the present note we begin a response to this question by extending the work of [6] to complexes associated with forms of arbitrary positive degree. In a sequel we shall similarly extend the results of multiplicity in [7].

The multiplicity of a set of homogeneous polynomials over a commutative ring

Some thirty years ago Buchsbaum and Rim [1] extended the notion of multiplicity e(a1, …, an; E) for elements a1, …, an of a commutative ring R with identity and a Noetherian R-module E (≠0) with lengthR (E/[sum ]nt=1aiE) finite to give a multiplicity e((aij); E) associated with E and an m×n matrix (aij) over R satisfying a certain extended finiteness condition. One of their results states that for each of a set of m complexes depending on E, (aij) the Euler–Poincaré characteristic is a certain integer multiple of e((aij); E), at least when R is a local ring.Some of these ideas were taken up in [4] where it is shown that when n[ges ]m−1, each of the complexes K((aij); E; t) with t∈ℤ introduced in [3] also have e((aij); E) as their Euler–Poincaré characteristic. With a slight change in viewpoint (aij) can be replaced by linear forms aj=[sum ]mi=1aijxi (j=1, …, n) of the graded polynomial ring R[x1, …, xm]; the complex K((aij); E; t) then becomes the component of degree t in a certain graded double complexformula herewhere K(a1, …, an; F) is the standard Koszul complex (see [4; section 2]). From this point of view the construction can be extended to allow the homogeneous polynomials a1, …, an to have any (possibly unequal) positive degrees [6]. The main aim of the present note is to extend similarly the results of [4] and to strengthen those results to give information on the vanishing of the multiplicity.

Algebraic invariants of the O(2) gauge transformation

The authors consider the O(2) gauge transformation for a two-state vertex model on a lattice, and derive its fundamental algebraic invariants, the minimal set of homogeneous polynomials of the vertex weights which are invariant under O(2) transformations. Explicit expressions of the fundamental invariants are given for symmetric vertex models on lattices with coordination number p=2, 3, 4, 5, 6, generalising p=3 results obtained previously from more elaborate considerations.