On the number of generators of a projective module

In this article we give a bound on the number of generators of a finitely generated projective module of constant rank over a commutative Noetherian ring in terms of the rank of the module and the dimension of the ring. Under certain conditions we provide an improvement to the Forster–Swan bound in case of finitely generated projective modules of rank n over an affine algebra over a finite field or an algebraically closed field.     

A bound on the plurigenera of projective varieties

We exhibit a sharp Castelnuovo bound for the i-th plurigenus of a smooth variety of given dimension n and degree d in the projective space P ^r , and classify the varieties attaining the bound, when n2, r2n+1, d>>r and i>>r. When n=2 and r=5, or n=3 and r=7, we give a complete classification, i.e. for any i1. In certain cases, the varieties with maximal plurigenus are not Castelnuovo varieties, i.e. varieties with maximal geometric genus. For example, a Castelnuovo variety complete intersection on a variety of dimension n+1 and minimal degree in P ^r , with r>(n ² +3n)/(n–1), has not maximal i-th plurigenus, for i>>r. As a consequence of the bound on the plurigenera, we obtain an upper bound for the self-intersection of the canonical bundle of a smooth projective variety, whose canonical bundle is big and nef. Mathematics Subject Classification (2000):Primary 14J99; Secondary 14N99     

An upper bound for the expected number of real zeros of a random polynomial

Suppose that the n + 1 coefficients of a polynomial of degree n are independent normally distributed random variables with mean 0 and variance 1. We prove that the expected number of real zeros of this polynomial is less than $\text{(}\text{2}\text{π}\text{)}\text{log}\text{n + 1.116}$, and that the constant may be reduced to 1.113 if n is even.     

A bound on the chromatic number of a graph

We give an upper bound on the chromatic number of a graph in terms of its maximum degree and the size of the largest complete subgraph. Our results extends a theorem due to Brooks.     

A lower bound for the minimum weight of the dual 7-ary code of a projective plane of order 49

Existing bounds on the minimum weight d ^⊥ of the dual 7-ary code of a projective plane of order 49 show that this must be in the range 76 ≤ d ^⊥ ≤ 98. We use combinatorial arguments to improve this range to 88 ≤ d ^⊥ ≤ 98, noting that the upper bound can be taken to be 91 if the plane has a Baer subplane, as in the desarguesian case. A brief survey of known results for the minimum weight of the dual codes of finite projective planes is also included. Dedicated to Dan Hughes on the occasion of his 80th birthday.     

A tight bound on the projective dimension of four quadrics

Motivated by Stillman's question, we show that the projective dimension of an ideal generated by four quadric forms in a polynomial ring is at most 6; moreover, this bound is tight. We achieve this bound, in part, by giving a characterization of the low degree generators of ideals primary to height three primes of multiplicities one and two.     

Existence of unimodular elements in a projective module

(1) Let R be an affine algebra over an algebraically closed field of characteristic 0 with $\dim (R)=n$. Let P be a projective $A=R[T_1,?,T_k]$-module of rank n with determinant L. Suppose I is an ideal of A of height n such that there are two surjections $\alpha :P?I$ and $?:L\oplus A^{n?1}?I$. Assume that either (a) $k=1$ and $n\ge 3$ or (b) k is arbitrary but $n\ge 4$ is even. Then P has a unimodular element (see 4.1, 4.3).(2) Let R be a ring containing $\mathbb{Q}$ of even dimension n with height of the Jacobson radical of $R\ge 2$. Let P be a projective $R[T,T^{?1}]$-module of rank n with trivial determinant. Assume that there exists a surjection $\alpha :P?I$, where $I?R[T,T^{?1}]$ is an ideal of height n such that I is generated by n elements. Then P has a unimodular element (see 3.4).     

A polynomial time upper bound for the number of contacts in the HP-model on the face-centered-cubic lattice (FCC)

Lattice protein models are a major tool for investigating principles of protein folding. For this purpose, one needs an algorithm that is guaranteed to find the minimal energy conformation in some lattice model (at least for some sequences). So far, there are only algorithm that can find optimal conformations in the cubic lattice. In the more interesting case of the face-centered-cubic lattice (FCC), which is more protein-like, there are no results. One of the reasons is that for finding optimal conformations, one usually applies a branch-and-bound technique, and there are no reasonable bounds known for the FCC. We will give such a bound for Dill's HP-model on the FCC, which can be calculated by a dynamic programming approach.     

A bound on the order of jumping lines

Mathematische Annalen -     

A remark on the average number of divisors of a quadratic polynomial

Archiv der Mathematik - In recent work, we use Dudek’s method together with a result of Zagier to establish an asymptotic formula for the average number of divisors of an irreducible...     

An upper bound on the number of cuts needed in Gomory's method of integer forms

The cut discipline of Gomory's first finiteness proof for the method of integer forms is used to derive an upper bound on the number of cuts required to obtain solution to pure integer programs.     

A sharp bound for the number of points where a rank 3 stable reflexive sheaf is not free

Let ³ be the projective space over an algebraically closed field k and let E be a rank 3 stable reflexive sheaf on ³ such that its restriction to a general plane is stable. The aim of this paper is to give a sharp bound of s= lengthExt ¹ (E,G)in terms of the Chern classes of E.     

On the number of generators needed for free profinite products of finite groups

We provide lower estimates on the minimal number of generators of the profinite completion of free products of finite groups. In particular, we show that if C ₁, ..., C _n are finite cyclic groups then there exists a finite group G which is generated by isomorphic copies of C ₁, ..., C _n and the minimal number of generators of G is n. The first author’s research is partially supported by NSF grant DMS-0701105. The second author’s research is partially supported by OTKA grant T38059 and the Magyary Zoltán Postdoctoral Fellowship.