GROp

radical of the Gräffe operator

Calling sequence:

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

Parameters:

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

Description:

• The radical of the Gräffe operator is defined by the following property:
  • If $p(x)$ is an irreducible polynomial, the irreducible factors $q(x)$ of $\operatorname{GROp}(p(x), x, b)$ are the irreducible polynomials whose image by the Mahler operator $\operatorname{MOp}$ with radix $b$ are $p(x)$ and each irreducible factor $q(x)$ has multiplicity $1$ in $\operatorname{GROp}(p(x), x, b)$.
    
• In other words $\operatorname{GROp}$ acts as an inverse of $\operatorname{MOp}$ on the set of irreducible polynomials without introducing multiplicities.

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 highlight the difference between the Gräffe operator $\operatorname{GOp}$ and its radical $\operatorname{GROp}$.

 f := MOp(8*x - 1, x, 3);

$$8 x^{3}-1$$

 g1 := GOp(f, x, 3);

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

 factor(g1);

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

 g2 := GROp(f, x, 3);

$$8 x -1$$