RamRatPolyToLMOp¶

from a ramified rational function to a linear Mahler operator¶

Calling sequence:¶

$\operatorname{RamRatPolyToLMOp}(f, x, M, b)$

Parameters:¶

  • $f$, a ramified rational function
  • $x$, a name, the variable of $f$
  • $M$, the name of the Mahler operator
  • $b$, a positive integer
  • option '${\mathit output}$' = '$\mathit{polynomial}$' / '$\mathit{monic}$'

Description:¶

  • For a given ramified rational function $f$, $\operatorname{RamRatPolyToLMOp}(f, x, M, b)$ provides a linear Mahler operator $L$ which annihilates $f$ of minimal order.
  • In addition, among all these operators, the output is chosen to have polynomial coefficients and constant content with respect to $M$, when the option '${\mathit output}$' = '$\mathit{polynomial}$' is used . It is therefore unique up to a constant in the base field.
  • If the option '${\mathit output}$' = '$\mathit{monic}$' is used, then a polynomial monic wrt $M$ is returned. This constraint makes it unique. Obviously the coefficients can be rational functions in $x$.
  • Particularly, given a rational function, $\operatorname{RamRatPolyToLMOp}$ provides a binomial linear Mahler operator of order $1$ which annihilates on this rational function.

Example:¶

In [1]:
libname := libname, FileTools:-JoinPath(["maple","lib","dcfun.mla"],base=homedir):
In [2]:
 with(dcfun):
Out[2]:

For a rational function, an order $1$ operator does the job and is obviously of minimal order among all nonzero linear Mahler operators which annihilates the rational function.

In [3]:
 b := 2;
Out[3]:

$$2$$

In [4]:
 f := x/(1 + x);
Out[4]:

$$\frac{x}{1+x}$$

In [5]:
 L := RamRatPolyToLMOp(f, x, M, b);
Out[5]:

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

In [6]:
 LMOpEval(L, f, x, M, b);
Out[6]:

$$0$$

For a ramified rational function, the order of the provided operator may be greater than $1$.

In [7]:
 f := x^(1/3)/(1 + x);
Out[7]:

$$\frac{x^{\frac{1}{3}}}{1+x}$$

In [8]:
 L := RamRatPolyToLMOp(f, x, M, b);
Out[8]:

$$M^{2} \left(x^{4}+1\right)-x^{2}-x$$

In [9]:
 LMOpEval(L, f, x, M, b);
Out[9]:

$$0$$

With the option '${\mathit output}$' = '$\mathit{monic}$', the unique monic polynomial which annihilates on $f$ is provided.

In [10]:
 L := RamRatPolyToLMOp(f, x, M, b, 'output' = 'monic');
Out[10]:

$$M^{2}-\frac{x \left(1+x \right)}{x^{4}+1}$$