Group Gradings on Polynomial Algebras

Let k be a field. We consider gradings on a polynomial algebra k[X₁,…, X_n] by an arbitrary abelian group G, such that the indeterminates are homogeneous elements of nontrivial degree. We classify the isomorphism types of such gradings, and we count them in the case where G is finite. We present some examples of good gradings and find a minimal set of generators of the subalgebra of elements of trivial degree.     

On the complexity of the multivariate resultant

The multivariate resultant is a fundamental tool of computational algebraic geometry. It can in particular be used to decide whether a system of n$n$ homogeneous equations in n$n$ variables is satisfiable (the resultant is a polynomial in the system’s coefficients which vanishes if and only if the system is satisfiable). In this paper, we investigate the complexity of computing the multivariate resultant.     

A simple approach for determining the eigenvalues of the fourth-order Sturm–Liouville problem with variable coefficients

In this paper, a simple and efficient approach is presented to compute the eigenvalues of the fourth-order Sturm–Liouville equations with variable coefficients. By transforming the governing differential equation to a system of algebraic equation, we can get the corresponding polynomial characteristic equations for kinds of boundary conditions based on the polynomial expansion and integral technique. Moreover, the lower and higher-order eigenvalues can be determined simultaneously from the multi-roots. Several examples for estimating eigenvalues are given. The convergence and effectiveness of the method are confirmed by comparing numerical results with the exact and other existing numerical results.     

Reduced Gröbner Bases of Certain Toric Varieties; A New Short Proof

We prove that every additively-idempotent semiring can be embedded in a finitary complete semiring. From this we obtain, among other results, that the classical identities of Kleene semirings over idempotent semirings are independent.     

Parametrizing Over ℤ Integral Values of Polynomials Over ℚ

Given a polynomial f ∈ ?[X] such that f(?) ? ?, we investigate whether the set f(?) can be parametrized by a multivariate polynomial with integer coefficients, that is, the existence of g ∈ ?[X ₁,…, X _m ] such that f(?) = g(? ^m ). We offer a necessary and sufficient condition on f for this to be possible. In particular, it turns out that some power of 2 is a common denominator of the coefficients of f, and there exists a rational β with odd numerator and odd prime-power denominator such that f(X) = f(β ?X). Moreover, if f(?) is likewise parametrizable, then this can be done by a polynomial in one or two variables.     

Twisted Planes

Let k be a commutative ring. We find and characterize a new family of twisted planes (i.e., associative unitary k-algebra structures on the k-module k[X, Y], having k[X] and k[Y] as subalgebras). Similar results are obtained for the k-module of two variables power series k[[X, Y]].     

On Finite Basis Property for Joins of Varieties of Associative Rings

If all finitely generated rings in a variety of associative rings satisfy the ascending chain condition on two-sided ideals, the variety is called locally weak noetherian. If there is an upper bound on nilpotency indices of nilpotent rings in a variety, the variety is called a finite index variety. We prove that the join of a finitely based locally weak noetherian variety and a variety of finite index is also finitely based and locally weak noetherian. One consequence of this result is that if an associative ring variety is connected by a finite path in the lattice of all associative ring varieties to a finitely based locally weak noetherian variety then such variety is also finitely based and locally weak noetherian.     

Binomial Canonical Decompositions of Binomial Ideals

In this article, we prove that every binomial ideal in a polynomial ring over an algebraically closed field of characteristic zero admits a canonical primary decomposition into binomial ideals. Moreover, we prove that this special decomposition is obtained from a cellular decomposition which is also defined in a canonical way and does not depend on the field.