Abc Theorem Research Articles

Generalization of the ABC Theorem on Locally Nilpotent Derivations

Generalization of the ABC Theorem on Locally Nilpotent Derivations

On the Rigidity of Some Extensions of Domains

A ring is said to be rigid if it admits no nonzero locally nilpotent derivations, and an affine variety is rigid if its coordinate ring is rigid. In this paper, we improve some techniques for determining the rigidity of k-domains (affine varieties) over a field k of characteristic zero. First, we generalize the ABC theorem. Then we study locally nilpotent derivations of a simple algebraic extension R[z] of a k-domain R, where rzn∈R for some nonzero r∈R and some positive integer n. Subsequently, we study locally nilpotent derivations and rigidity of an extension R[x,y] of R such that r1xmyn∈R or r1xm+r2yn∈R for some nonzero r1,r2∈R and some positive integers m,n. Finally, as applications of these general results, we prove the rigidity of some quadrinomial varieties and Pham–Brieskorn hypersurfaces.

A Stothers\u2013Mason theorem with a difference radical

Differential calculus is not a unique way to observe polynomial equations such as a+b=c. We propose a way of applying difference calculus to estimate multiplicities of the roots of the polynomials a, b and c satisfying the equation above. Then a difference abc theorem for polynomials is proved using a new notion of a radical of a polynomial. Results, for example, on the non-existence of polynomial solutions to difference Fermat and difference Super-Fermat functional equations are given as applications. We also introduce a truncated second main theorem for differences, and use it to consider these functional equations with non-polynomial entire solutions. Equations with polynomial or non-polynomial solutions are observed to see the sharpness of results obtained.

Multiple orthogonal polynomials of mixed type: Gauss–Borel factorization and the multi-component 2D Toda hierarchy

Multiple orthogonality is considered in the realm of a Gauss–Borel factorization problem for a semi-infinite moment matrix. Perfect combinations of weights and a finite Borel measure are constructed in terms of M-Nikishin systems. These perfect combinations ensure that the problem of mixed multiple orthogonality has a unique solution, that can be obtained from the solution of a Gauss–Borel factorization problem for a semi-infinite matrix, which plays the role of a moment matrix. This leads to sequences of multiple orthogonal polynomials, their duals and second kind functions. It also gives the corresponding linear forms that are bi-orthogonal to the dual linear forms. Expressions for these objects in terms of determinants from the moment matrix are given, recursion relations are found, which imply a multi-diagonal Jacobi type matrix with snake shape, and results like the ABC theorem or the Christoffel–Darboux formula are re-derived in this context (using the factorization problem and the generalized Hankel symmetry of the moment matrix). The connection between this description of multiple orthogonality and the multi-component 2D Toda hierarchy, which can be also understood and studied through a Gauss–Borel factorization problem, is discussed. Deformations of the weights, natural for M-Nikishin systems, are considered and the correspondence with solutions to the integrable hierarchy, represented as a collection of Lax equations, is explored. Corresponding Lax and Zakharov–Shabat matrices as well as wave functions and their adjoints are determined. The construction of discrete flows is discussed in terms of Miwa transformations which involve Darboux transformations for the multiple orthogonality conditions. The bilinear equations are derived and the τ-function representation of the multiple orthogonality is given.

An abc theorem on the disk

We extend the classical abc theorem for polynomials (also known as Mason's, or Mason–Stothers', theorem) to general analytic functions on the disk.

Generalized ABC theorems for non-Archimedean entire functions of several variables in arbitrary characteristic

We prove generalized ABC theorems for vanishing sums of non-Archimedean entire functions of several variables in arbitrary characteristic.

WITHDRAWN: An abc theorem for arbitrarily many polynomials

WITHDRAWN: An abc theorem for arbitrarily many polynomials

The ABC theorem for higher-dimensional function fields

We generalize the ABC theorems to the function field of a variety over an algebraically closed field of arbitrary characteristic which is non-singular in codimension one. We also obtain an upper bound for the minimal order sequence of Wronskians over such function fields of positive characteristic.

A note on Wronskians and the ABC theorem in function fields of prime characteristic

We provide a sharp bound for the order sequence of Wronskians. We also give another proof of the truncated second main theorem over function fields which is a generalization of the ABC theorem due to Mason, Voloch, Brownawell and Masser, Noguchi and the author.

The abc theorem for commutative algebraic groups in characteristic p

Journal Article The abc theorem for commutative algebraic groups in characteristic p Get access Thomas Scanlon Thomas Scanlon Search for other works by this author on: Oxford Academic Google Scholar International Mathematics Research Notices, Volume 1997, Issue 18, 1997, Pages 881–898, https://doi.org/10.1155/S1073792897000573 Published: 01 January 1997 Article history Published: 01 January 1997 Received: 23 May 1997 Revision received: 23 July 1997