# Documentation

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# Cat::Polynomial

Category of multivariate polynomials

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## Syntax

```Cat::Polynomial(`R`)
```

## Description

`Cat::Polynomial(R)` represents the category of multivariate polynomials over `R`.

A `Cat::Polynomial(R)` is a multivariate polynomial ring over a commutative coefficient ring `R`.

## Categories

If `R` is a `Cat::FactorialDomain`, then `Cat::FactorialDomain`.

If `R` is a `Cat::GcdDomain`, then `Cat::GcdDomain`.

If `R` is a `Cat::IntegralDomain`, then `Cat::IntegralDomain`.

## Parameters

 `R` A domain which must be from the category `Cat::CommutativeRing`.

## Entries

 "coeffRing" The coefficient ring `R`. "characteristic" The characteristic of this domain, which is the same as that of the ring `R`.

## Methods

expand all

#### Basic Methods

`coeff(p)`

`coeff(p, x, n)`

`coeff(p, n)`

Must return the coefficient of `x^n` of `p`, which is a polynomial in the remaining indeterminates.

Must return the coefficient of `x^n` of `p`, where `x` is the main variable of `p`.

`degree(p)`

`degree(p, x)`

Must return the degree of `p` with respect to the indeterminate `x`.

`degreevec(p)`

`evalp(p, x = v, …)`

More than one evaluation point may be given. The result must be a polynomial in the remaining indeterminates or an element of `R`.

`indets(p)`

`lcoeff(p)`

`lmonomial(p)`

`lterm(p)`

`mainvar(p)`

```mapcoeffs(p, f, <a, …>)```

`multcoeffs(p, c)`

`nterms(p)`

`nthcoeff(p, n)`

`nthmonomial(p, n)`

`nthterm(p, n)`

`tcoeff(p)`

`unitNormal(p)`

An implementation is provided if `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(p)`

An implementation is provided if `R` has the axiom `Ax::canonicalUnitNormal`.

#### Mathematical Methods

`content(p)`

`isUnit(p)`

`primpart(p)`

`poly2list(p)`

```solve(p, x, <opt, …>)```

```solve(p, x = T, <opt, …>)```

`solve(p)`

Solves the polynomial equation `p = 0` with respect to `x` over the domain `T`. See the function `solve` for details about the optional arguments `opt, ...`

The polynomial `p` must be univariate. Solves the polynomial equation `p = 0` with respect to the indeterminate of `p` over the domain `R`.