## `lterm` -- the leading term of a polynomial

#### Introduction

`lterm(`p`)` returns the leading term of the polynomial `p`.

#### Call(s)

`lterm(p <, vars> <, order>)`

#### Parameters

`p` | - | a polynomial of type `DOM_POLY` or a polynomial expression
`vars` | - | a list of indeterminates of the polynomial: typically, identifiers or indexed identifiers
`order` | - | the term ordering: either `LexOrder` or `DegreeOrder` or `DegInvLexOrder` or a user-defined term ordering of type `Dom::MonomOrdering`. The default is the lexicographical ordering LexOrder.

#### Returns

a polynomial of the same type as `p`. An expression is returned if `p` is an expression. `FAIL` is returned if the input cannot be converted to a polynomial.

### Overloadable:

`p`

#### Related Functions

`coeff`, `degree`, `degreevec`, `ground`, `lcoeff`, `ldegree`, `lmonomial`, `nterms`, `nthcoeff`, `nthmonomial`, `nthterm`, `poly`, `poly2list`, `tcoeff`

#### Details

- The argument `p` can either be a polynomial expression, or a polynomial generated by `poly`, or an element of some polynomial domain overloading `lterm`.
- The identity lterm(p) * lcoeff(p) = lmonomial(p) holds.
- If a list of indeterminates is provided, then `p` is regarded as a polynomial in these indeterminates. Note that the specified list does not have to coincide with the indeterminates of the input polynomial. Cf. example 1.
- The returned term is ``leading'' with respect to the lexicographical ordering, unless a different ordering is specified via the argument `order`. Cf. example 2.
- `lterm` returns `FAIL` if the input polynomial cannot be converted to a polynomial in the specified indeterminates. Cf. example 3.
- The leading term of the zero polynomial is the zero polynomial.
- For the orderings LexOrder, DegreeOrder and DegInvLexOrder, the result is computed by a fast kernel function. Other orderings are handled by slower library functions.

#### Example 1

We demonstrate how the indeterminates influence the result:

    >> p := 2*x^2*y + 3*x*y^2 + 6:
       lterm(p), lterm(p, [x, y]), lterm(p, [y, x])

                                    2   2       2
                                 x y , x  y, x y

Note that the indeterminates passed to `lterm` will be used, even if the polynomial provides different indeterminates :

    >> p := poly(2*x^2*y + 3*x*y^2, [x, y]):
       lterm(p), lterm(p, [x, y]), lterm(p, [y, x]),
       lterm(p, [y]), lterm(p, [z])

                2                   2                   2
          poly(x  y, [x, y]), poly(x  y, [x, y]), poly(y  x, [y, x]),

                   2
             poly(y , [y]), poly(1, [z])

    >> delete p:

#### Example 2

We demonstrate how various orderings influence the result:

    >> p := poly(5*x^4 + 4*x^3*y*z^2 + 3*x^2*y^3*z + 2, [x, y, z]):
       lterm(p), lterm(p, DegreeOrder), lterm(p, DegInvLexOrder)

                4                    3    2
          poly(x , [x, y, z]), poly(x  y z , [x, y, z]),

                   2  3
             poly(x  y  z, [x, y, z])

The following call uses the reverse lexicographical order on 3 indeterminates:

    >> lterm(p, Dom::MonomOrdering(RevLex(3)))

                                   2  3
                             poly(x  y  z, [x, y, z])

    >> delete p:

#### Example 3

We define a polynomial over the integers modulo 7:

    >> p := poly(3*x + 4, [x], Dom::IntegerMod(7)): lterm(p)

                         poly(x, [x], Dom::IntegerMod(7))

This polynomial cannot be regarded as a polynomial with respect to another indeterminate, because the ``coefficient'' `x` cannot be interpreted as an element of the coefficient ring `Dom::IntegerMod``(7)`:

    >> lterm(p, [y])

                                       FAIL

    >> delete p:

#### Example 4

The leading monomial is the product of the leading coefficient and the leading term:

    >> p := poly(2*x^2*y + 3*x*y^2 + 6, [x, y]):
       mapcoeffs(lterm(p),lcoeff(p)) = lmonomial(p)

                            2                      2
                    poly(2 x  y, [x, y]) = poly(2 x  y, [x, y])

    >> delete p:

#### Changes

- Now it is possible to specify user defined term orderings.
- Indeterminates can now be specified for polynomials of type `DOM_POLY` as well.
- In previous MuPAD releases, `lterm` was a kernel function.

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