Journal of Algebra and Its Applications

This paper contains a complete and self-contained proof of the AMS theorem. The main tool is a new algorithm producing an irreducible dependence for a pair of polynomials in one variable.

Teissier has proven remarkable inequalities for intersection numbers s_i = (ℒⁱ ⋅ ℳ^d-i) of a pair of nef line bundles ℒ, ℳ on a d-dimensional complete algebraic variety over a field. He asks if two nef and big line bundles are numerically proportional if the inequalities are all equalities. In this paper, we show that this is true in the most general possible situation, for nef and big line bundles on a proper irreducible scheme over an arbitrary field k. Boucksom, Favre and Jonsson have recently established this result on a complete variety X over an algebraically closed field of characteristic zero. Their proof involves an ingenious extension of the intersection theory on a variety to its Zariski Riemann Manifold. This extension requires the existence of a direct system of nonsingular varieties dominating X. We make use of a simpler intersection theory which does not require resolution of singularities, and extend volume to an arbitrary field and prove its continuous differentiability, extending results of Boucksom, Favre and Jonsson, and of Lazarsfeld and Mustaţă. A goal in this paper is to provide a manuscript which will be accessible to many readers. As such, subtle topological arguments which are required to give a complete proof in [S. Bouksom et al., J. Algebraic Geometry18 (2009) 279–308] have been written out in this manuscript, in the context of our intersection theory, and over arbitrary varieties.

The problem of finite generation of the kernel of a derivation of a polynomial ring is a special case of Hilbert's Fourteenth Problem. It is well known that the answer is affirmative if the derivation is locally nilpotent and having a slice. Van den Essen (1995) conjectured that there exists a counterexample for non-locally nilpotent derivations with a slice. In this paper, we solve this conjecture in the affirmative.

The abundancy of a whole number n is the ratio of the sum of divisors of n to n; for instance, all perfect numbers have abundancy two. Two numbers are called friends if they have the same abundancy; a number is said to be friendly if it has at least one friend and solitary if it has no friends. Here we find several new families of solitary numbers.

This paper is a survey of two subjects: the first part is devoted to field generators in two variables, and the second to birational endomorphisms of the affine plane. Each one of these subjects originated in Abhyankar's seminar in Purdue University in the 1970s. Note that the part on field generators is more than a survey, since it contains a considerable amount of new material.

We prove some interesting multiplicative relations which hold between the coefficients of the logarithmic derivatives obtained in a few simple ways from 𝔽_q-linear formal power series. Since the logarithmic derivatives connect power sums to elementary symmetric functions via the Newton identities, we establish, as applications, new identities between important quantities of function field arithmetic, such as the Bernoulli–Carlitz fractions and power sums as well as their multivariable generalizations. Resulting understanding of their factorizations has arithmetic significance, as well as applications to function field zeta and multizeta values evaluations and relations between them. Using specialization/generalization arguments, we provide much more general identities on linear forms providing a switch between power sums for positive and negative powers.

In [On affine-ruled rational surfaces, Math. Ann.255(3) (1981) 287–302], Russell had proved that when k is a perfect field of positive characteristic, the polynomial ring k[X, Y] is cancellative. In this note, we shall show that this cancellation property holds even without the hypothesis that k is perfect.

We establish some results about orbit closures of permutation groups and their relationship to the automorphism groups of power-set graphs with colorings. One of our theorems proves that an arbitrary group can be realized as the automorphism group of a colored power-set graph.

In this work we deal with dicritical divisors, curvettes and polynomials. These objects have been one of the main research interests of Abhyankar during his last years. In this work we provide some elementary proofs of some Abhyankar and Luengo results for dicriticals in the framework of formal power series. Based on these ideas we give a constructive way to find the atypical fibers of a special pencil and give bounds for its number, which are sharper than the existing ones. Finally, we answer a question of Gwoździewicz finding polynomials that reach his bound.

The celebrated "Epimorphism Theorem" opened up several directions of research. In this survey article we highlight the known partial results and open questions on (i) the epimorphism problem for higher dimensions and (ii) the generalized epimorphism problems over nice integral domains (when the corresponding results over fields are known). We shall also discuss a few results and questions on Affine Fibrations, the Cancellation Problem and the Linearization Problem which are closely related to the epimorphism (or embedding) problems.

Our purpose is to study the family of simple undirected graphs whose toric ideal is a complete intersection from both an algorithmic and a combinatorial point of view. We obtain a polynomial time algorithm that, given a graph G, checks whether its toric ideal P_G is a complete intersection or not. Whenever P_G is a complete intersection, the algorithm also returns a minimal set of generators of P_G. Moreover, we prove that if G is a connected graph and P_G is a complete intersection, then there exist two induced subgraphs R and C of G such that the vertex set V(G) of G is the disjoint union of V(R) and V(C), where R is a bipartite ring graph and C is either the empty graph, an odd primitive cycle, or consists of two odd primitive cycles properly connected. Finally, if R is 2-connected and C is connected, we list the families of graphs whose toric ideals are complete intersection.

Using the theory of minimal models of quasi-projective surfaces we give a new proof of the theorem of Lin–Zaidenberg which says that every topologically contractible algebraic curve in the complex affine plane has equation Xⁿ = Y^m in some algebraic coordinates on the plane. This gives also a proof of the theorem of Abhyankar–Moh–Suzuki concerning embeddings of the complex line into the plane. Independently, we show how to deduce the latter theorem from basic properties of ℚ-acyclic surfaces.

We give explicit parametric equations for all irreducible plane projective sextic curves which have at most double points and whose total Milnor number is maximal (is equal to 19). In each case we find a parametrization over a number field of the minimal possible degree and try to choose coordinates so that the coefficients are as small as we can do.

We solve the problem of effectively computing the a-invariant of ladder determinantal rings. In the case of a one-sided ladder, we provide a compact formula, while, for a large family of two-sided ladders, we provide an algorithmic solution.