The proof of the following theorem is presented: If D is, respectively, a Krull domain, a Dedekind domain, or a PrGfer domain, then D is correspondingly a UFD, a PID, or a Bezout domain if and only if every irreducible quadratic polynomial in D[X] is a prime element.

This paper is concerned with the parametrization of all the decentralized stabilizing controllers. The auxiliary diagonal system, which is defined by the diagonal elements of Bezout factors, plays important roles in the parametrization of decentralized controllers. This paper gives an explicit characterization of the auxiliary system.

We solve the problem of robust stabilization with respect to rightcoprime factor perturbations for irrational discrete-time transfer functions. The key condition is that the associated dynamical system and its dual should satisfy a finite-cost condition so that two optimal cost operators exist. We obtain explicit state space formulas for a robustly stabilizing controller in terms of these optim...

Let A be a commutative ring and E a non-zero A-module. Necessary and sufficient conditions are given for the trivial ring extension R of A by E to be either arithmetical or Gaussian. The possibility for R to be Bézout is also studied, but a response is only given in the case where pSpec(A) (a quotient space of Spec(A)) is totally disconnected. Trivial ring extensions which are fqp-rings are cha...

A foundation polynomial is used to induce polynomial bases for F n?1 x], the vector space of polynomials of degree less than n over an arbitrary eld F. The associated bases are then used to block diagonalize the Bezout matrix of two polynomials under congruence.

5 Divisors 30 5.1 The Picard group . . . . . . . . . . . . . . . . . . . . . . . . . 30 5.2 Automorphisms of Pnk and of A n k . . . . . . . . . . . . . . . . 33 5.3 The degree of the divisor of a rational function on a projective curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 5.4 Bezout theorem for curves . . . . . . . . . . . . . . . . . . . . 36 5.5 Riemann–Roch theo...

The Bezout-Dixon resultant method for solving systems of polynomial equations lends itselfto various heuristic acceleration techniques, previously reported by the present author [6], which can beextraordinarily effective. In this paper we will discuss how well these techniques apply to the Macaulayresultant. In brief, we find that they do work there with some difficulties, but the D...

The resultant is an algebraic expression, computable in a finite number of arithmetic operations from the coefficients of two univariate polynomials, that vanishes if, and only if, the two polynomials have common zeros. The paper considers formal resultant for degree-deficient polynomials (polynomials whose actual degree is lower than their assumed degree). Some key properties of the resultant ...