Subsections

• Related Structures
• Numerical Invariants
• Ring Predicates and Booleans

The main structure related to a polynomial ring is its coefficient ring. Multivariate polynomial rings belong to the Magma category RngDPol.

Category(P) : RngDPol -> Cat

Parent(P) : RngDPol -> Pow

PrimeRing(P) : RngDPol -> Rng

CoefficientRing(P) : RngDPol -> Rng

Return the coefficient ring of polynomial ring P.

Note that the # operator only returns a value for finite (quotients of) polynomial rings.

Characteristic(P) : RngDPol -> RngIntElt

# P : RngDPol -> RngIntElt

Return the number of indeterminates of polynomial ring P over its coefficient ring.

The usual ring functions returning Boolean values are available on polynomial rings. There is also a function that checks if a multivariate polynomial is in fact univariate in a given variable.

IsCommutative(P) : RngDPol -> BoolElt

IsUnitary(P) : RngDPol -> BoolElt

IsFinite(P) : RngDPol -> BoolElt

IsOrdered(P) : RngDPol -> BoolElt

IsField(P) : RngDPol -> BoolElt

IsEuclideanDomain(P) : RngDPol -> BoolElt

IsPID(P) : RngDPol -> BoolElt

IsUFD(P) : RngDPol -> BoolElt

IsDivisionRing(P) : RngDPol -> BoolElt

IsEuclideanRing(P) : RngDPol -> BoolElt

IsPrincipalIdealRing(P) : RngDPol -> BoolElt

IsDomain(P) : RngDPol -> BoolElt

P eq Q : RngDPol, RngDPol -> BoolElt

P ne Q : RngDPol, RngDPol -> BoolElt

Given a multivariate polynomial p in R[x_1, ..., x_n], this function returns true if and only if p is in fact a univariate polynomial in one of its indeterminates x_1, ..., x_n. If true is returned, then the function also returns a univariate version q of p and (the first) i such that p is univariate in x_i. Note that there will only be ambiguity about i if p is a constant polynomial. The univariate polynomial q will be an element of R[x] with the same coefficients as p.

IsUnivariate(p, i) : RngDPolElt, RngIntElt -> BoolElt, RngUPolElt

IsUnivariate(p, v) : RngDPolElt, RngUPolElt -> BoolElt, RngUPolElt

Given a multivariate polynomial p in R[x_1, ..., x_n], this function returns true if and only if p is in fact a univariate polynomial in x_i. If true is returned, then the function also returns a univariate version q of p, which will be an element of the univariate polynomial ring R[x] with the same coefficients as p. The indeterminate x_i should either be specified as a (polynomial) argument v or as an integer i.