This library implements functional uni-multivariate decomposition
of multivariate polynomials.
A (multivariate) polynomial f is a composite if it can be written as
986#986 where g is univariate and h is multivariate,
where
987#987.
Uniqueness for monic polynomials is up to linear coordinate change
988#988.
If f is a composite, then decompose(f);
returns an ideal (g,h);
such that
989#989 is maximal, (
990#990).
The polynomial h is, by the maximality of
991#991, not a composite.
The polynomial g is univariate in the (first) variable vvar of f,
such that deg_vvar(f) is maximal.
decompose(f,1);
computes a full decomposition, i.e. if f is a
composite, then an ideal
992#992 is returned, where
993#993 are univariate and each entry is primitive such that
994#994.
If f is not a composite, for instance if
995#995 is prime,
then decompose(f);
returns f.
The command decompose
is the inverse: compose(decompose(f,1))==f
.
Recall, that Chebyshev polynomials of the first kind commute by composition.
The decomposition algorithms work in the tame case, that is if
char(basering)=0 or p:=char(basering) > 0 but deg(g) is not divisible by
p.
Additionally, it works for monic polynomials over
30#30 and in some
cases for monic polyomials over coefficient rings.
See
is_composite
for examples. (It also works over the reals but
there it seems not be numerical stable.)
More information on the univariate resp. multivariate case.
Univariate decomposition is created, with the additional assumption
987#987.
A multivariate polynomial f is a composite, if f can be written as
986#986, where
149#149 is a univariate polynomial and
996#996
is multivariate. Note, that unlike in the univariate case, the polynomial
996#996 may be of degree
294#294.
E.g.
997#997 is the composite of
998#998 and
999#999.
If nvars(basering)>1
, then, by default, a single-variable
multivariate polynomial is not considered to be the same as in the
one-variable polynomial ring; it will always be decomposed. That is:
> ring r1=0,x,dp;
> decompose(x3+2x+1);
x3+2x+1
but:
> ring r2=0,(x,y),dp;
> decompose(x3+2x+1);
_[1]=x3+2x+1
_[2]=x
In particular:
is_composite(x3+2x+1)==1;
in ring r1
but
is_composite(x3+2x+1)==0;
in ring r2
.
This is justified by interpreting the polynomial decomposition as an
affine Stein factorization of the mapping
1000#1000.
The behaviour can changed by the some global variables.
int DECMETH;
choose von zur Gathen's or Kozen-Landau's method.
int MINS;
compute f = g o h, such that h(0) = 0.
int IMPROVE;
simplify the coefficients of g and h if f is
not monic.
int DEGONE;
single-variable multivariate are
considered uni-variate.
See decompopts;
for more information.
Additional information is displayed if printlevel > 0
.