Algebra of polynomials bounded on some strips

Abstract

In the present paper, for a finite sequence of single variable polynomials g (x) = (g1(x), g2(x), ..., gn(x)), we study the ring of geometrically bounded elements on a generalized strip Mc(g) in ℝn+1 which is the solution of the system of polynomial inequalities g1(x) £ y1 £ g1(x) + c1, g2(x) £ y2 £ g2(x) + c2 …, gn(x) £ yn £ gn(x) + cn. This ring is shown to be the finitely generated ℝ-algebra ℝ[y1 - g1(x), y2 - g2(x), ..., yn - gn(x)] provided that c = (c1, c2, ..., cn) is a positive vector. However, if c = 0 then this algebra is not finitely generated in general. In particular, we point out that the ring of geometrically bounded elements on ‘a generalized strip’ of the form M(g1, g2) in ℝ2 which is the solution of the polynomial inequality g1(x) £ y £ g2(x) is trivial (i.e., is equal to ℝ) provided that g1(x) is less than g2(x) at infinity. As a consequence, we can describe the ring of geometrically bounded elements on a finite union of disjoint strips