The Shorter function determines if f is shorter than g, where "shorter" means:

smaller length

monic in x

no fractions

lower degree in other variables

The Sort function sorts a list of polynomials in x according to the ordering defined by Shorter.

The Shorten function applies Mobius transformations to  f in an effort to make f shorter (in the above sense). More precisely, the following transformations are tried: $x\mapsto \frac{a\cdot x+b}{c\cdot x+d}$ whenever has([a, b, c, d], x)=false and $\det \left(\left[\left[a\,b\right]\,\left[c\,d\right]\right]\right)\ne 0$. Furthermore, a polynomial f may be divided by lcoeff(f, x) or by content(f, x), or by any expression that does not contain x.

When a field extension is given by an irreducible polynomial f, these procedures can be used to search for a simpler polynomial representing the same field extension.

If f contains only one variable, then the second argument x in Shorten is optional.

This function is part of the PolynomialTools package, and so it can be used in the form Shorten(..) only after executing the command with(PolynomialTools). However, it can always be accessed through the long form of the command by using PolynomialTools[Shorten](..).

$\mathrm{with}\left(\mathrm{PolynomialTools}\right)\:$

$\mathrm{Shorten}\left(x^2+x+1\,x\right)$

$x^2+3$

$\mathrm{Shorten}\left(3x^3+18x+14\,x\right)$

$x^3-6$

$\mathrm{Shorten}\left(x^4+32\right)$

$x^4+2$

$\mathrm{Shorter}\left(x^3\,x+5\,x\right)$

$\mathrm{false}$

$\mathrm{Sort}\left(\left[x^3\,x^2\,x+1\,x+5\right]\,x\right)$

$\left[1+x\,x+5\,x^2\,x^3\right]$