# Documentation

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# `nthmonomial`

N-th monomial of a polynomial

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## Syntax

```nthmonomial(`p`, `n`)
nthmonomial(`f`, <`vars`>, `n`)
```

## Description

`nthmonomial(p, n)` returns the `n`-th non-trivial monomial of the polynomial `p`.

`nthmonomial` returns the `n`-th non-trivial monomial with respect to the lexicographical ordering.

The “first” monomial is the leading monomial as returned by `lmonomial`.

A zero polynomial has no terms: `nthmonomial` returns `FAIL`.

A polynomial expression `f` is first converted to a polynomial with the variables given by `vars`. If no variables are given, they are searched for in `f`. See `poly` about details of the conversion. The result is returned as polynomial expression. `FAIL` is returned if `f` cannot be converted to a polynomial.

The result of `nthmonomial` is not fully evaluated. It can be evaluated by the functions `mapcoeffs` and `eval`. Cf. Example 2.

## Examples

### Example 1

We give some self explaining examples:

```p := poly(100*x^100 + 49*x^49 + 7*x^7, [x]): nthmonomial(p, 1), nthmonomial(p, 2), nthmonomial(p, 3)```

`nthmonomial(p, 4)`

`nthmonomial(poly(0, [x]), 1)`

`delete p:`

### Example 2

We demonstrate the evaluation strategy of `nthmonomial`:

```p := poly(3*x^3 + 6*x^2*y^2 + 2, [x]): y := 4: nthmonomial(p, 2)```

Evaluation is enforced by `eval`:

`mapcoeffs(%, eval)`

`delete p, y:`

## Parameters

 `p` A polynomial of type `DOM_POLY` `f` `vars` A list of indeterminates of the polynomial: typically, identifiers or indexed identifiers `n` A positive integer

## Return Values

Polynomial of the same type as `p`. An expression is returned if a polynomial expression is given as input. `FAIL` is returned if `n` is larger than the actual number of terms of the polynomial.

`p`