All rings are commutative with identity and all modules are unital. Let R be a ring, M an R-module and R (M), the idealization of M . Homogeneous ideals of R (M) have the form I (+)N where I is an ideal of R, N a submodule of M and IM ⊆ N . The purpose of this paper is to investigate how properties of a homogeneous ideal I (+)N of R (M) are related to those of I and N . We show that if M is a m...

An algebraic branching program (ABP) is a directed acyclic graph, with a start vertex s, and end vertex t and each edge having a weight which is an affine form in F[x1, x2, . . . , xn]. An ABP computes a polynomial in a natural way, as the sum of weights of all paths from s to t, where the weight of a path is the product of the weights of the edges in the path. An ABP is said to be homogeneous ...

In this note we discuss an analog of the classical Waring problem for C[x0,x1,...,x(n)]. Namely, we show that a general homogeneous polynomial p ∈ C[x0,x1,...,x(n)] of degree divisible by k≥2 can be represented as a sum of at most k(n) k-th powers of homogeneous polynomials in C[x0,x1,...,x(n)]. Noticeably, k(n) coincides with the number obtained by naive dimension count.

This paper proposes a new construction of the visual secret sharing scheme for the (n, n)-threshold access structure applicable to color images. The construction uses matrices with n rows that can be identified with homogeneous polynomials of degree n. It is shown that, if we find a set of homogeneous polynomials of degree n satisfying a certain system of simultaneous partial differential equat...

We prove some simple properties of the power series ring R = Kx 1 ; x 2 ; x 3 ; : : : ]] on a countably innnite number of variables over a eld K, and of the subring R 0 generated by all homogeneous elements in R. By means of a certain decreasing ltration of ideals, which are kernels of the \truncation homomorphisms" n : R 0 ! Kx 1 ; : : : ; x n ], we endow R 0 with a topology, and show that wit...

The goal of this article is to provide a construction of a homogeneous Lyapunov function P associated with a system of differential equations J = f(x), x ~ R ~ (n > 1), under the hypotheses: (1) f ~ C(R n, ~ ) vanishes at x = 0 and is homogeneous; (2) the zero solution of this system is locally asymptotically stable. Moreover, the Lyapunov function V(x) tends to infinity with 1[ x [I, and belon...