This function reduces powers of algebraic numbers or algebraic functions modulo their defining polynomials.

Algebraic functions and algebraic numbers may be represented by radicals or with the RootOf notation (see type,algnum, type,algfun, type,radnum, type,radfun).

The result will have the form P/Q, where P and Q are polynomials over the extension field. The powers of the RootOfs appearing in the coefficients are positive and lower than the degree of the defining polynomials.

By default, this function attempts to preserve partial factorization of polynomials, but algebraic numbers and functions are always expanded. If the option expanded is specified, then polynomials are also expanded.

Unlike evala@Normal, the function Reduce does not rationalize algebraic numbers and functions and does not rationalize leading coefficients of rational functions and polynomials. It also does not cancel the greatest common divisor of the numerator and the denominator of a rational function.

If the RootOfs appearing in the input are independent, then this function will return 0 if and only if the input is mathematically equal to 0. It may not be so if the RootOfs are dependent or if the polynomial defining a RootOf is reducible.

If a contains functions, their arguments are reduced recursively and the functions are frozen before the computation proceeds.

Since the ordering of objects may vary from a session to another, the result may change accordingly.

Other objects are frozen and considered as variables, except in the cases below.

If a is a set, a list, a range, a relation, or a series, then Reduce is mapped over the object.

$\mathrm{r1}≔\mathrm{RootOf}\left({\mathrm{\_Z}}^3+\mathrm{\_Z}+1\right)$

$\mathrm{r1}≔\mathrm{RootOf}\left({\mathrm{\_Z}}^3+\mathrm{\_Z}+1\right)$

$\mathrm{p1}≔\frac{{\left(x-{\mathrm{r1}}^3-1\right)}^2}{\left({\mathrm{r1}}^3+1\right){\left(x-\mathrm{r1}\right)}^2}$

$\mathrm{p1}≔\frac{{\left(-{\mathrm{RootOf}\left({\mathrm{\_Z}}^3+\mathrm{\_Z}+1\right)}^3+x-1\right)}^2}{\left({\mathrm{RootOf}\left({\mathrm{\_Z}}^3+\mathrm{\_Z}+1\right)}^3+1\right){\left(x-\mathrm{RootOf}\left({\mathrm{\_Z}}^3+\mathrm{\_Z}+1\right)\right)}^2}$

$\mathrm{evala}\left(\mathrm{Reduce}\left(\mathrm{p1}\right)\right)$

$-\frac{{\left(\mathrm{RootOf}\left({\mathrm{\_Z}}^3+\mathrm{\_Z}+1\right)+x\right)}^2}{{\left(-x+\mathrm{RootOf}\left({\mathrm{\_Z}}^3+\mathrm{\_Z}+1\right)\right)}^2\mathrm{RootOf}\left({\mathrm{\_Z}}^3+\mathrm{\_Z}+1\right)}$

$\mathrm{evala}\left(\mathrm{Reduce}\left(\mathrm{p1}\right)\,\mathrm{expanded}\right)$

$\frac{{\mathrm{RootOf}\left({\mathrm{\_Z}}^3+\mathrm{\_Z}+1\right)}^2+x^2+2x\mathrm{RootOf}\left({\mathrm{\_Z}}^3+\mathrm{\_Z}+1\right)}{\mathrm{RootOf}\left({\mathrm{\_Z}}^3+\mathrm{\_Z}+1\right)+1+2{\mathrm{RootOf}\left({\mathrm{\_Z}}^3+\mathrm{\_Z}+1\right)}^{2}x-\mathrm{RootOf}\left({\mathrm{\_Z}}^3+\mathrm{\_Z}+1\right)x^2}$

$\mathrm{r2}≔\mathrm{RootOf}\left({\mathrm{\_Z}}^2-y\right)$

$\mathrm{r2}≔\mathrm{RootOf}\left({\mathrm{\_Z}}^2-y\right)$

$\mathrm{evala}\left(\mathrm{Reduce}\left(\frac{t-\mathrm{r2}}{t^2-y}\right)\right)$

$-\frac{-t+\mathrm{RootOf}\left({\mathrm{\_Z}}^2-y\right)}{t^2-y}$

$\mathrm{r3}≔\mathrm{RootOf}\left({\mathrm{\_Z}}^2-4\right)$

$\mathrm{r3}≔\mathrm{RootOf}\left({\mathrm{\_Z}}^2-4\right)$

$\mathrm{evala}\left(\mathrm{Reduce}\left({\mathrm{r3}}^3-8\right)\right)$

$4\mathrm{RootOf}\left({\mathrm{\_Z}}^2-4\right)-8$

$\mathrm{alias}\left(\mathrm{seq}\left(s\Vert i=\mathrm{RootOf}\left({\mathrm{\_Z}}^3-2\,\mathrm{index}=i\right)\,i=1..3\right)\right)$

$\mathrm{s1},\mathrm{s2},\mathrm{s3}$

$q≔{\mathrm{s1}}^4{\mathrm{s2}}^3{\mathrm{s3}}^2$

$q≔{\mathrm{s1}}^4{\mathrm{s2}}^3{\mathrm{s3}}^2$

$\mathrm{evala}\left(\mathrm{Reduce}\left(q\right)\right)$

$4\mathrm{s1}{\mathrm{s3}}^2$