Length Four Polynomial Automorphisms

We study the structure of length four polynomial automorphisms of R[X, Y] when R is a unique factorization domain. The results from this study are used to prove that, if SL _m (R[X ₁, X ₂,…, X _n ]) = E _m (R[X ₁, X ₂,…, X _n ]) for all n, m ≥ 0, then all length four polynomial automorphisms of R[X, Y] that are commutators are stably tame.     

On questions related to saturated chains of primes in polynomial rings

We propose to give positive answers to the open questions: is R(X,Y) strong S when R(X) is strong S? is R stably strong S (resp., universally catenary) when R[X] is strong S (resp., catenary)? in case R is obtained by a (T,I,D) construction. The importance of these results is due to the fact that this type of ring is the principal source of counterexamples. Moreover, we give an answer to the open questions: is R〈X₁,…,X_n〉 residually Jaffard (resp., totally Jaffard) when R(X₁,…,X_n) is ? We construct a three-dimensional local ring R such that R(X₁,…,X_n) is totally Jaffard (and hence, residually Jaffard) whereas R〈X₁,…,X_n〉 is not residually Jaffard (and hence, not totally Jaffard).     

Bitableau bases for Garsia-Haiman modules of hollow type

Garsia-Haiman modules C[Xn,Yn]/Iγ are quotient rings in the variables Xn={x₁,x₂,…,xn} and Yn={y₁,y₂,…,yn} that generalize the quotient ring C[Xn]/I, where I is the ideal generated by the elementary symmetric polynomials ej(Xn) for 1?j?n. A bitableau basis for the Garsia-Haiman modules of hollow type is constructed. Applications of this basis to representation theory and other related polynomial spaces are considered.     

On the modular sumset partition problem

A sequence m₁≥m₂≥?≥m_k of k positive integers isn-realizable if there is a partition X₁,X₂,…,X_k of the integer interval [1,n] such that the sum of the elements in X_i is m_i for each i=1,2,…,k. We consider the modular version of the problem and, by using the polynomial method by Alon (1999) [2], we prove that all sequences in Z/pZ of length k≤(p−1)/2 are realizable for any prime p≥3. The bound on k is best possible. An extension of this result is applied to give two results of p-realizable sequences in the integers. The first one is an extension, for n a prime, of the best known sufficient condition for n-realizability. The second one shows that, for n≥(4k)³, an n-feasible sequence of length k isn-realizable if and only if it does not contain forbidden subsequences of elements smaller than n, a natural obstruction forn-realizability.     

Conditional ordering of order statistics

For any positive integers m and n, let X₁,X₂,…,X_m∨n be independent random variables with possibly nonidentical distributions. Let X_1:n≤X_2:n≤?≤X_n:n be order statistics of random variables X₁,X₂,…,X_n, and let X_1:m≤X_2:m≤?≤X_m:m be order statistics of random variables X₁,X₂,…,X_m. It is shown that (X_j:n,X_j+1:n,…,X_n:n) given X_i:m>y for j−i≥max{n−m,0}, and (X_1:n,X_2:n,…,X_j:n) given X_i:m≤y for j−i≤min{n−m,0} are all increasing in y with respect to the usual multivariate stochastic order. We thus extend the main results in Dubhashi and Häggström (2008) [1] and Hu and Chen (2008) [2].     

Stochastic comparisons of order statistics from gamma distributions

Let (X₁,X₂,…,X_n) and (Y₁,Y₂,…,Y_n) be gamma random vectors with common shape parameter α(0<α?1) and scale parameters (λ₁,λ₂,…,λ_n), (μ₁,μ₂,…,μ_n), respectively. Let X₍₎=(X₍₁₎,X₍₂₎,…,X_(n)), Y₍₎=(Y₍₁₎,Y₍₂₎,…,Y_(n)) be the order statistics of (X₁,X₂,…,X_n) and (Y₁,Y₂,…,Y_n). Then (λ₁,λ₂,…,λ_n) majorizes (μ₁,μ₂,…,μ_n) implies that X₍₎ is stochastically larger than Y₍₎. However if the common shape parameter α>1, we can only compare the the first- and last-order statistics. Some earlier results on stochastically comparing proportional hazard functions are shown to be special cases of our results.     

Cyclic irreducible non-holonomic modules over the Weyl algebra: An algorithmic characterization

Let K be a field of characteristic zero, n≥1 an integer and A_n+1=K[X,Y₁,…,Y_n]〈∂_X,∂_Y1,…,∂_Yn〉 the (n+1)th Weyl algebra over K. Let S∈A_n+1 be an order-1 differential operator of the type with a_i,b_i∈K[X] and g_i∈K[X,Y_i] for every i=1,…,n. We construct an algorithm that allows one to recognize whether S generates a maximal left ideal of A_n+1, hence also whether A_n+1/A_n+1S is an irreducible non-holonomic A_n+1-module. The algorithm, which is a powerful instrument for producing concrete examples of cyclic maximal left ideals of A_n, is easy to implement and quite useful; we use it to solve several open questions.The algorithm also allows one to recognize whether certain families of algebraic differential equations have a solution in K[X,Y₁,…,Y_n] and, when they have one, to compute it.     

On asymptotic distributions of exceedance statistics

In this study, two independent samples X₁,X₂,…,X_n and Y₁,Y₂,…,Y_m with respective distribution functions F and Q are considered. The joint asymptotic distributions of exceedance statistics defined as the number of Y observations falling into a random interval of order statistics constructed from the X sample is investigated. The results can be used in the context of a two-sample problem.     

On the range of heterogeneous samples

Let R_n be the range of a random sample X₁,…,X_n of exponential random variables with hazard rate λ. Let S_n be the range of another collection Y₁,…,Y_n of mutually independent exponential random variables with hazard rates λ₁,…,λ_n whose average is λ. Finally, let r and s denote the reversed hazard rates of R_n and S_n, respectively. It is shown here that the mapping t?s(t)/r(t) is increasing on (0,∞) and that as a result, R_n=X_(n)−X₍₁₎ is smaller than S_n=Y_(n)−Y₍₁₎ in the likelihood ratio ordering as well as in the dispersive ordering. As a further consequence of this fact, X_(n) is seen to be more stochastically increasing in X₍₁₎ than Y_(n) is in Y₍₁₎. In other words, the pair (X₍₁₎,X_(n)) is more dependent than the pair (Y₍₁₎,Y_(n)) in the monotone regression dependence ordering. The latter finding extends readily to the more general context where X₁,…,X_n form a random sample from a continuous distribution while Y₁,…,Y_n are mutually independent lifetimes with proportional hazard rates.     

Derivations of polynomial algebras without Darboux polynomials

We present several new examples of homogeneous derivations of a polynomial ring k[X]=k[x₁,…,x_n] over a field k of characteristic zero without Darboux polynomials. Using a modification of a result of Shamsuddin, we produce these examples by induction on the number n of variables, thus more easily than the previously known example multidimensional Jouanolou systems of ?o?a?dek.     

Generalizations of a lemma of Freudenburg

Let k be an algebraically closed field of characteristic zero and ℘ a prime ideal in k[X]?k[X₁,…,X_n]. Let g∈k[X] and d?1. If for all 1?|α|?d the derivatives ∂^αg belong to ℘, then there exists c∈k such that g−c∈℘(d+1), the d+1th symbolic power of ℘. In particular, if ℘ is a complete intersection it follows that g−c∈℘d+1.     

On the diophantine equation x(x + 1)(x + 2)…(x + (m − 1)) =g(y)

Let g(y) ? Q[Y] be an irreducible polynomial of degree n ≥ 3. We prove that there are only finitely many rational numbers x, y with bounded denominator and an integer m ≥ 3 satisfying the equation x(x + 1) (x + 2)…(x + (m − 1) ) = g(y). We also obtain certain finiteness results when g(y) is not an irreducible polynomial.     

The intersection number of real polynomial mappings

Let X=V(f₁,…,fn−m)⊂Rn be a compact real algebraic set and g:X→R2m be a continuous function. If the diagonal in X×X is isolated in the set of self-intersection points of g, we define the intersection number of g. In the case where X is a manifold and g is an immersion it is the intersection number defined by Whitney. In the case where g is a polynomial mapping, we present an effective formula for this number.     

On the dependence between the extreme order statistics in the proportional hazards model

Let X₁,…,X_n be a random sample from an absolutely continuous distribution with non-negative support, and let Y₁,…,Y_n be mutually independent lifetimes with proportional hazard rates. Let also X₍₁₎<?<X_(n) and Y₍₁₎<?<Y_(n) be their associated order statistics. It is shown that the pair (X₍₁₎,X_(n)) is then more dependent than the pair (Y₍₁₎,Y_(n)), in the sense of the right-tail increasing ordering of Avérous and Dortet-Bernadet [LTD and RTI dependence orderings, Canad. J. Statist. 28 (2000) 151-157]. Elementary consequences of this fact are highlighted.     

On universally catenarian pairs

If 1≤n<∞ and R⊆S are integral domains, then (R,S) is called an n-catenarian pair if for each intermediate ring T (that is each ring T such that R⊆T⊆S) the polynomial ring in n indeterminates, T[n] is catenarian. This implies that (R,S) is m-catenarian for all m<n. The main purpose of this paper is to prove that 1-catenarian and universally catenarian pairs are equivalent in several cases. An example of a 1-catenarian pair which is not 2-catenarian is given.     

A study of the lex plus powers conjecture

An {a₁,…,a_n}-lex plus powers ideal is a monomial ideal in I⊂k[x₁,…,x_n] which minimally contains the regular sequence x₁^a₁,…,x_n^a_n and such that whenever m∈R_t is a minimal generator of I and m′∈R_t is greater than m in lex order, then m′∈I. Conjectures of Eisenbud et al. and Charalambous and Evans predict that after restricting to ideals containing a regular sequence in degrees {a₁,…,a_n}, then {a₁,…,a_n}-lex plus powers ideals have extremal properties similar to those of the lex ideal. That is, it is proposed that a lex plus powers ideal should give maximum possible Hilbert function growth (Eisenbud et al.), and, after fixing a Hilbert function, that the Betti numbers of a lex plus powers ideal should be uniquely largest (Charalambous, Evans). The first of these assertions would extend Macaulay's theorem on Hilbert function growth, while the second improves the Bigatti, Hulett, Pardue theorem that lex ideals have largest graded Betti numbers. In this paper we explore these two conjectures. First we give several equivalent forms of each statement. For example, we demonstrate that the conjecture for Hilbert functions is equivalent to the statement that for a given Hilbert function, lex plus powers ideals have the most minimal generators in each degree. We use this result to prove that it is enough to show that lex plus powers ideals have the most minimal generators in the highest possible degree. A similar result holds for the stronger conjecture. In this paper we also prove that if the weaker conjecture holds, then lex plus powers ideals are guaranteed to have largest socles. This suffices to show that the two conjectures are equivalent in dimension ?3, which proves the monomial case of the conjecture for Betti numbers in those degrees. In dimension 2, we prove both conjectures outright.     

Derivations and right ideals of algebras

Let R be a K-algebra acting densely on V_D, where K is a commutative ring with unity and V is a right vector space over a division K-algebra D. Let ρ be a nonzero right ideal of R and let f(X₁,…,X_t) be a nonzero polynomial over K with constant term 0 such that μR≠0 for some coefficient μ of f(X₁,…,X_t). Suppose that d:R→R is a nonzero derivation. It is proved that if rankd(f(x₁,…,x_t))?m for all x₁,…,x_t∈ρ and for some positive integer m, then either ρ is generated by an idempotent of finite rank or d=ad(b) for some b∈End(V_D) of finite rank. In addition, if f(X₁,…,X_t) is multilinear, then b can be chosen such that rank(b)?2(6t+13)m+2.     

Polynomials over ordered fields

In this paper we determine which polynomials over ordered fields have multiples with nonnegative coefficients and also which polynomials can be written as quotients of two polynomials with nonnegative coefficients. This problem is related to a result given by Pólya in [G.H. Hardy, J.E. Littlewood, G. Pólya, Inequalities, Cambridge University Press, Cambridge, England, 1952] (as a companion of Artin’s theorem) that asserts that if F(X₁,…,X_n)∈R[X₁,…,X_n] is a form (i.e., a homogeneous polynomial) s.t.  with ∑x_j>0, then F=G/H, where G,H are forms with all coefficients positive (i.e., every monomial of degree degG or degH appears in G or H, resp., with a coefficient that is strictly positive). In Pólya’s proof H is chosen to be H=(X₁+?+X_n)^m for some m.At the end we give some applications, including a generalization of Pólya’s result to arbitrary ordered fields.     

Note on bounds for multiplicities

Let S=K[x₁,…,x_n] be a polynomial ring and R=S/I be a graded K-algebra where I⊂S is a graded ideal. Herzog, Huneke and Srinivasan have conjectured that the multiplicity of R is bounded above by a function of the maximal shifts in the minimal graded free resolution of R over S. We prove the conjecture in the case that codim(R)=2 which generalizes results in (J. Pure Appl. Algebra 182 (2003) 201; Trans. Amer. Math. Soc. 350 (1998) 2879). We also give a proof for the bound in the case in which I is componentwise linear. For example, stable and squarefree stable ideals belong to this class of ideals.     

The inversion formulae for automorphisms of polynomial algebras and rings of differential operators in prime characteristic

Let K be an arbitrary field of characteristic p>0. We find an explicit formula for the inverse of any algebra automorphism of any of the following algebras: the polynomial algebra P_n?K[x₁,…,x_n], the ring of differential operators D(P_n) on P_n, D(P_n)⊗P_m, the n’th Weyl algebra A_n or A_n⊗P_m, the power series algebra K[[x₁,…,x_n]], the subalgebra T_k1,…,kn of D(P_n) generated by P_n and the higher derivations , 0≤j<p^k_i, i=1,…,n (where k₁,…,k_n∈N), T_k1,…,kn⊗P_m or an arbitrary central simple (countably generated) algebra over an arbitrary field.