GOp

Gräffe operator

Calling sequence:

$\operatorname{GOp}(p, x, b)$

Parameters:

• $p$ a polynomial in $x$
  
• $x$, a name
  
• $b$, a positive integer

Description:

• The Gräffe operator, here implemeted by the procedure $\operatorname{GOp}$, is defined by the formula $\operatorname{GOp}(p(x), x, b) = \operatorname{Res}_y( y^b - x, p(y))$, where $\operatorname{Res}_y$ is the resultant wrt $y$.
  
• It appears as a quasi-inverse of the Mahler operator in view of the relations
  • $\operatorname{GOp}(\operatorname{MOp}(p(x), x, b), x, b) = p(x)^b$,
    
  • $p(x)$ divides $\operatorname{MOp}(\operatorname{GOp}(p(x), x, b), x, b)$.
    
• If $p(x)$ is an irreducible polynomial, the irreducible factors of $\operatorname{GOp}(p(x), x, b)$ are the irreducible polynomials whose images by $\operatorname{MOp}$ are $p(x)$.

References:

• Frédéric Chyzak, Thomas Dreyfus, Philippe Dumas, and Marc Mezzarobba (2018). Computing solutions of linear Mahler equations. Mathematics of Computation 87.314, pp. 2977–3021.

Example:

libname := libname, FileTools:-JoinPath(["maple","lib","dcfun.mla"],base=homedir):

 with(dcfun):

We illustrate the formula $p \mid \operatorname{MOp}(\operatorname{GOp}(p, x, b), x, b)$.

 p := 8*x - 1;

$$8 x -1$$

 MG := MOp(GOp(p, x, 3), x, 3);

$$512 x^{3}-1$$

 factor(MG);

$$\left(8 x -1\right) \left(64 x^{2}+8 x +1\right)$$

We illustrate the formula $\operatorname{GOp}(\operatorname{MOp}(p, x, b), x, b) = p^b$.

 GM := GOp(MOp(p, x, 3), x, 3);

$$512 x^{3}-192 x^{2}+24 x -1$$

 factor(GM);

$$\left(8 x -1\right)^{3}$$