A conjugation-invariant ideal I ⊆ C[zj , zj : j = 1, . . . , n] has the Quillen property if every real valued, strictly positive polynomial on the real zero set VR(I) ⊆ C n is a sum of hermitian squares modulo I . We first relate the Quillen property to the archimedean property from real algebra. Using hereditary calculus, we then quantize and show that the Quillen property implies the subnorma...

Definition 14.1. Let X ⊆ Am and Y ⊆ An be affine varieties. A morphism f : X → Y is a ̄ map f(P ) := (f1(P ), . . . , fn(P )) defined by polynomials f1, . . . , fn ∈ k[x1, . . . , xm] such that f(P ) ∈ Y for all points P ∈ X. We may regard f1, . . . , fn as representatives of elements of ̄ ̄ the coordinate ring k[X] = k[x1, . . . , xm]/I(X); we are evaluating f1, . . . , fn only at points in X, so...

We introduce circulant matrices that capture the structure of a skew-polynomial ring F[x; θ] modulo the left ideal generated by a polynomial of the type x − a. This allows us to develop an approach to skew-constacyclic codes based on such circulants. Properties of these circulants are derived, and in particular it is shown that the transpose of a certain circulant is a circulant again. This rec...

LetM =M(E) be a matroid on a linear ordered set E . The Orlik–Solomon Z-algebra OS(M) of M is the free exterior Z-algebra on E , modulo the ideal generated by the circuit boundaries. The Z-module OS(M) has a canonical basis called ‘no broken circuit basis’ and denoted nbc. Let eX = ∏ ei , ei ∈ X ⊂ E . We prove that when eX is expressed in the nbc basis, then all the coefficients are 0 or ±1. We...

Given n polynomials in n variables of respective degrees d1, . . . , dn, and a set of monomials of cardinality d1 . . . dn, we give an explicit subresultant-based polynomial expression in the coefficients of the input polynomials whose non-vanishing is a necessary and sufficient condition for this set of monomials to be a basis of the ring of polynomials in n variables modulo the ideal generate...

We study a form of refined class number formula (resp. type formula) for maximal orders in totally definite quaternion algebras over real quadratic fields, by taking into consideration the automorphism groups right ideal classes unit orders). For each finite noncyclic group $G$, we give an explicit conjugacy whose modulo center are isomorphic to and write down representative class. This leads c...

The free abelian group R(Q) on the set of indecomposable representations of a quiver Q, over a field K, has a ring structure where the multiplication is given by the tensor product. We show that if Q is a rooted tree (an oriented tree with a unique sink), then the ring R(Q)red is a finitely generated Z-module (here R(Q)red is the ring R(Q) modulo the ideal of all nilpotent elements). We will de...

In this paper, we present another tree-structure signature scheme based on discrete logarithm problem modulo p, where p is a large prime. The basic signing algorithm is the original ELGmal signature scheme. The scheme attains ideal security, i.e, finding existential forgeries under adaptively chosen message attacks is equivalent to solving the discrete logarithm of any random integer y ∈ Z∗ p ....

Let K be a finite tamely ramified extension of Qp and let L/K be a totally ramified (Z/pnZ)-extension. Let πL be a uniformizer for L, let σ be a generator for Gal(L/K), and let f(X) be an element of ØK [X] such that σ(πL) = f(πL). We show that the reduction of f(X) modulo the maximal ideal of ØK determines a certain subextension of L/K up to isomorphism. We use this result to study the field ex...