Calculation of the dimensional polynomial of a prime principal difference ideal (Q1076075)

Let F be a difference field of characteristic zero with base set of commuting automorphisms \(\Delta =\{\alpha,\beta \}\) and \(F\{\) \(y\}\) be the ring of difference polynomials. Each \(A\in F\{y\}\) is placed in correspondence with a finite set \(\tau\) (A) of integer-valued vectors of plane \({\mathbb{R}}^ 2\) as follows: (p,q)\(\in \tau (A)\) if and only if the element \(\alpha^ p\beta^ qy\) is present in A. The set \(\tau\) (A) is bounded by the minimal rectangle, for which the normals to the sides are equal to \((\pm 1,\pm 1)\), its semiperimeter is called the difference order of A. It is shown that the high-order coefficient of the dimensional polynomial of a prime principal difference ideal is equal to the difference order of the polynomial that generates it.