RamificationOrder¶

Calling sequence:¶

$\operatorname{RamificationOrder}(f, v)$

Parameters:¶

  • $f$, expression
  • $v$, variable, i.e. a name or symbol, or a list or set of such names or symbols

Description:¶

  • $\operatorname{RamificationOrder}(f, v)$ does not returns $\mathit FAIL$ iff $f$ is a ramified rational function of $v$. Then it returns the minimal suitable ramification order for $f$.
  • The returned ramification order $q$ is an integer or a list of integers in the special case where $v$ is a list of variables.

Example:¶

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

We consider a ramified polynomial.

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

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

In [4]:
 RamificationOrder(f, x);
Out[4]:

$$6$$

The following expression is not a ramified rational function because of the exponential term.

In [5]:
 f := x + sqrt(x) + x^(1/3) + exp(x);
Out[5]:

$$x +\sqrt{x}+x^{\frac{1}{3}}+{\mathrm e}^{x}$$

In [6]:
 RamificationOrder(f, x);
Out[6]:

$$\mathit{FAIL}$$

But note that $\operatorname{RamificationOrder}$ only considers the expression in relation to the variable(s) given as the second argument.

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

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

In [8]:
 RamificationOrder(f, x);
Out[8]:

$$2$$

In [9]:
 f := x + sqrt(x) + y^(1/3) + exp(y);
Out[9]:

$$x +\sqrt{x}+y^{\frac{1}{3}}+{\mathrm e}^{y}$$

In [10]:
 RamificationOrder(f, x);
Out[10]:

$$2$$

In [11]:
 f := x + sqrt(x) + y^(1/3);
Out[11]:

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

In [12]:
 RamificationOrder(f, {x, y});
Out[12]:

$$6$$

If a list of variables is given as second argument, a list of corresponding ramification orders is returned.

In [13]:
 f := x + sqrt(x) + y^(1/3);
Out[13]:

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

In [14]:
 RamificationOrder(f, [x, y]);
Out[14]:

$$[2, 3]$$