Cat::Polynomial -- the
category of multivariate polynomials
Introduction represents the category
of multivariate polynomials over Cat::Polynomial(R)R.
Generating
the categoryCat::Polynomial(R)
ParametersR |
- | A domain which must be from the category Cat::CommutativeRing. |
Cat::PartialDifferentialRing
, Cat::Algebra(R)
DetailsCat::Polynomial(R) is a multivariate
polynomial over a commutative coefficient ring R.The coefficient ring R.
The characteristic of this domain, which is is the same as that of
the ring R.
coeff(dom p)p.coeff(dom p, indeterminate x, Type::NonNegInt
n)x^n of p,
which is a polynomial in the remaining indeterminates.coeff(dom p, Type::NonNegInt n)x^n of p,
where x is the main variable of p.degree(dom p)p.degree(dom p, indeterminate x)p with respect to the
indeterminate x.degreevec(dom p)p. The order of the exponents corresponds to the order of
the indeterminates as given by the method "indets".evalp(dom p, indeterminate x = R v...)p at the point x =
v where x is an indeterminate and
v an element of R.R.indets(dom p)p.lcoeff(dom p)p.lmonomial(dom p)p.lterm(dom p)p.mainvar(dom p)p, which is the first
of the indeterminates as given by the method
"indets".mapcoeffs(dom p, function f <, a...>)c_i of p by
the results of the function calls f(c_i, a...).multcoeffs(dom p, R
c)p by
c.nterms(dom p)p.nthcoeff(dom p, Type::PosInt n)n-th coefficient of
p.nthmonomial(dom p, Type::PosInt n)n-th monomial of p.nthterm(dom p, Type::PosInt n)n-th term of p.tcoeff(dom p)p.unitNormal(dom p)p.R has the axiom
Ax::canonicalUnitNormal: In
this case p is multiplied by an unit of R
such that the leading coefficient has unit normal representation in
R.unitNormalRep(dom p)p and
the factors needed to bring p into unit normal form (see
Cat::IntegralDomain for the
return value expected).R has the axiom
Ax::canonicalUnitNormal.content(dom p)p if R is a
Cat::GcdDomain.isUnit(dom p)TRUE iff
p is a unit.primpart(dom p)p if R is a
Cat::GcdDomain: The
content of p is removed and the unit normal of the result
is returned.solve(dom p, indeterminate x <, opt...>)p = 0 with respect to
x over the domain R. See the function
solve for details about
the optional arguments opt....solve(dom p, indeterminate x = DOM_DOMAIN T <,
opt...>)p = 0 with respect to
x over the domain T. See the function
solve for details about
the optional arguments opt....solve(dom p)p must be univariate. Solves the
polynomial equation p = 0 with respect to the
indeterminate of p over the domain R.Cat::PolynomialCat.