Cayley-Bacharach Theorem Research Articles

F. S. Macaulay: From plane curves to Gorenstein rings

Francis Sowerby Macaulay began his career working on Brill and Noether’s theory of algebraic plane curves and their interpretation of the Riemann–Roch and Cayley–Bacharach theorems; in fact it is Macaulay who first stated and proved the modern form of the Cayley–Bacharach theorem. Later in his career Macaulay developed ideas and results that have become important in modern commutative algebra, such as the notions of unmixedness, perfection (the Cohen–Macaulay property), and super-perfection (the Gorenstein property), work that was appreciated by Emmy Noether and the people around her. He also discovered results that are now fundamental in the theory of linkage, but this work was forgotten and independently rediscovered much later. The name of a computer algebra program (now Macaulay2) recognizes that much of his work is based on examples created by refined computation. Though he never spoke of the connection, the threads of Macaulay’s work lead directly from the problems on plane curves to his later theorems. In this paper we will explain what Macaulay did, and how his results are connected.

A Cayley–Bacharach theorem and plane configurations

In this paper, we examine linear conditions on finite sets of points in projective space implied by the Cayley–Bacharach condition. In particular, by bounding the number of points satisfying the Cayley–Bacharach condition, we force them to lie on unions of low-dimensional linear spaces. These results are motivated by investigations into degrees of irrationality of complete intersections, which are controlled by minimum-degree rational maps to projective space. As an application of our main theorem, we describe the fibers of such maps for certain complete intersections of codimension two.

Geometric Methods in Partial Differential Equations

We study the interplay between geometry and partial differential equations. We show how the fundamental ideas we use require the ability to correctly calculate the dimensions of spaces associated to the varieties of zeros of the symbols of those differential equations. This brings to the center of the analysis several classical results from algebraic geometry, including the Cayley-Bacharach theorem and some of its variants as Serret’s theorem, and the Brill-Noether Restsatz theorem.

Sparse versions of the Cayley–Bacharach theorem

We give combinatorial generalizations of the Cayley–Bacharach theorem and induced map.

A Cayley–Bacharach theorem for points in Pn

We prove a Cayley-Bacharach-type theorem for points in projective space $\mathbb{P}^n$ that lie on a complete intersection of $n$ hypersurfaces. This is made possible by new bounds on the growth of the Hilbert function of almost complete intersections.

On the Noether and the Cayley–Bacharach Theorems with PD Multiplicities

In this paper we prove the Noether theorem with the multiplicities described by PD operators. Despite the known analog versions in this case the provided conditions are necessary and sufficient. We also prove the Cayley–Bacharach theorem with PD multiplicities. As far as we know this is the first generalization of this theorem for multiple intersections.

Virtual residue and a generalized Cayley-Bacharach theorem

Using the virtual residue, which is a generalization of the Grothendieck residue, we generalize the Cayley-Bacharach theorem to the cases with positive dimensions.

Extreme rays of Hankel spectrahedra for ternary forms

The cone of sums of squares is one of the central objects in convex algebraic geometry. Its defining linear inequalities correspond to the extreme rays of the dual convex cone. This dual cone is a spectrahedron, which can be explicitly realized as a section of the cone of positive semidefinite matrices with the linear subspace of Hankel (or middle catalecticant) matrices. In this paper we initiate a systematic study of the extreme rays of Hankel spectrahedra for ternary forms. We show that the Zariski closure of the union of extreme rays is the variety of all Hankel matrices of corank at least 4, an irreducible variety of codimension 10 and we determine its degree. We explicitly construct an extreme ray of maximal rank using the Cayley–Bacharach Theorem for plane curves. We apply our results to the study of the algebraic boundary of the cone of sums of squares. Its irreducible components are dual to varieties of Gorenstein ideals with certain Hilbert functions. We determine these Hilbert functions for some cases of small degree. We also observe surprising gaps in the ranks of Hankel matrices of the extreme rays.

Cayley–Bacharach Formulas

The Cayley–Bacharach theorem states that all cubic curves through eight given points in the plane also pass through a unique ninth point. We write that point as an explicit rational function in the other eight.

The Cayley-Bacharach theorem for continuous piecewise algebraic curves over cross-cut triangulations

A piecewise algebraic curve is a curve determined by the zero set of a bivariate spline function. In this paper, we propose the Cayley-Bacharach theorem for continuous piecewise algebraic curves over cross-cut triangulations. We show that, if two continuous piecewise algebraic curves of degrees m and n respectively meet at mnT distinct points over a cross-cut triangulation, where T denotes the number of cells of the triangulation, then any continuous piecewise algebraic curve of degree m + n − 2 containing all but one point of them also contains the last point.

Superposition interpolation process in [formula omitted

In this paper, we deeply research Lagrange interpolation by polynomial in several variables and give an application of Cayley–Bacharach theorem for it. Particularly on the sufficiently intersected algebraic manifold (or SIAM, for short), we introduce a general method of constructing properly posed set of nodes (or PPSN, for short) for Lagrange interpolation, namely the superposition interpolation process. Then we give an equivalent condition about a PPSN along a SIAM. Further we introduce a relation between the sufficiently intersected algebraic hypersurfaces and H-basis. At the end of this paper, we use the extended Cayley–Bacharach theorem to resolve some problems of Lagrange interpolation along the zero-dimensional and one-dimensional algebraic manifolds.

Two-variable polynomials: intersecting zeros and stability

In order to construct two-variable polynomials with a certain zero behavior, the notion of intersecting zeros is studied. We show that generically two-variable polynomials have a finite set of intersecting zeros, and give an algorithm on how to construct a polynomial with the desired intersecting zeros. Relations with the Cayley-Bacharach theorem are addressed. In addition, we will also address the case when stable polynomials are sought.

Some researches on trivariate Lagrange interpolation

In this paper, in order to go a step further research on the problem of trivariate Lagrange interpolation, we pose the concepts of sufficient intersection of algebraic surfaces and Lagrange interpolation along a space algebraic curve, and extend Cayley–Bacharach theorem in algebraic geometry from R 2 to R 3 . By using the conclusion of the extended theorem, we deduce a general method of constructing properly posed set of nodes for Lagrange interpolation along a space algebraic curve, and give a series of corollaries for the practical applications. Moreover, we give a new method of constructing properly posed set of nodes for Lagrange interpolation along an algebraic surface, and as a result we make clear the geometrical structure of it.

Cayley–Bacharach and evaluation codes on complete intersections

Hansen (Appl. Algebra Eng. Comm. Comput. 14 (2003) 175) uses cohomological methods to find a lower bound for the minimum distance of an evaluation code determined by a reduced complete intersection in P 2 . In this paper, we generalize Hansen's results from P 2 to P m ; we also show that the hypotheses of Hansen (2003) may be weakened. The proof is succinct and follows by combining the Cayley–Bacharach Theorem and the bounds on evaluation codes obtained in Hansen (Zero-Dimensional Schemes (Ravello, 1992), de Gruyter, Berlin, 1994, pp. 205–211).

Cayley–Bacharach theorem of piecewise algebraic curves

The piecewise algebraic curve, determined by a bivariate spline function, is a generalization of the classical algebraic curve. In this paper, by using Bezout's theorem and Nöther-type theorem of piecewise algebraic curves, the Cayley–Bacharach theorem and Hilbert function of C 0 piecewise algebraic curves are presented.

The application of Cayley–Bacharach Theorem to bivariate Lagrange interpolation

In this paper, we give a new proof of the famous Cayley–Bacharach theorem by means of interpolation, and deduce a general method of constructing properly posed set of nodes for bivariate Lagrange interpolation. As a result, we generalize the main results in Liang (On the interpolations and approximations in several variables, Jilin University, 1965), Liang and Lü (Approximation Theory IX, Vanderbilt University Press, 1988) and Liang et al. (Analysis, Combinatorics and Computing, Nova Science Publishers, Inc., New York, 2002) to the more extensive situations.

On the intersections of polynomials and the Cayley–Bacharach theorem

Let R= K[ x 1,…, x n ] and let f 1,…, f n be products of linear forms with f i of degree d i . Assume that the f i have d 1,…, d n common zeros. Then we determine the maximum number of those zeros that a form of degree k can go through without going through all of them. This is a version of a conjecture of Eisenbud, Green, and Harris. We suggest a possible method for using this to explore the case where the f i are arbitrary forms of degree d i with the right number of common zeros.

Theory of residues on the projective plane

An elementary theory of residues on the projective plane is given, which includes a canonical residual complex, a trace map, the residue theorem, and intersection numbers. Some classical theorems about plane projective geometry, such as the Reiss relation, the Cayley-Bacharach theorem, and the Newton theorem, are reproved via this approach of residues.

A homological proof of a theorem by Davis, Geramita, Orecchia giving the Cayley-Bacharach theorem.

A homological proof of a theorem by Davis, Geramita, Orecchia giving the Cayley-Bacharach theorem.

Gorenstein Algebras and the Cayley-Bacharach Theorem

This paper is an examination of the connection between the classical Cayley-Bacharach theorem for complete intersections in ${{\mathbf {P}}^2}$ and properties of graded Gorenstein algebras.