-- variable-reduced polynomials

Let f be an n-variate integer polynomial.

By range(f), we mean f(Z^n). Thus range(f) is a subset of Z.

Call integer polynomials f,g "range-equivalent" if range(f) =
range(g).

Call an n-variate integer polynomial f "variable reduced" if there
does not exist an m-variate integer polynomial g, range-equivalent to
f, such that m < n.

In other words, if f is variable reduced, then range(f) can't be
represented as the range of an integer polynomial with less variables.

Examples ...

Let

a = x
b = x + y
c = x1^2 + x2^2 + x3^2 + x4^2 - y1^2

Then a,b,c are range-equivalent to each other, but only a is
variable-reduced.

Here's a problem ...

Prove or disprove:

If integer polynomials f,g are range-equivalent, and if g is
variable-reduced, then f = g o h, for some integer polynomial h.

quasi
.

Relevant Pages

  • Re: -- variable-reduced polynomials
    ... Call integer polynomials f,g "range-equivalent" if ... If integer polynomials f,g are range-equivalent, and ...
    (sci.math)
  • Re: -- variable-reduced polynomials
    ... If integer polynomials f,g are range-equivalent, and if g is ... Prove or disprove: ... up to an invertible linear reparametrization. ...
    (sci.math)
  • Re: -- variable-reduced polynomials
    ... On 24.04.2008 08:23, quasi wrote: ... If integer polynomials f,g are range-equivalent, and if g is ... conjecture. ...
    (sci.math)
  • Re: -- variable-reduced polynomials
    ... On 24.04.2008 10:39, Jannick Asmus wrote: ... If integer polynomials f,g are range-equivalent, and if g is ...
    (sci.math)
  • Re: -- variable-reduced polynomials
    ... On 24.04.2008 06:22, quasi wrote: ... If integer polynomials f,g are range-equivalent, and if g is ... I think it is worth to remark that you are talking of a polynomial map h with range in Z^m for some m. m might be _strictly_ greater than 1. ...
    (sci.math)
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