If output=list (the default), a list of ordinals, nonnegative integers and polynomials with positive integer coefficients is returned.

Otherwise, if output=inert is specified, an inert product of ordinal numbers using the inert multiplication and exponentiation operators &. and &^, respectively, is returned. Factors equal to $1$ are omitted from this product representation.

The Factor(a) calling sequence computes a factored normal form of $a$ as a product of nonnegative integers and ordinals of the form ${\mathbf{\omega }}^d$ or ${\mathbf{\omega }}^d\+1$.

If $a\={\mathbf{\omega }}^{e_1}\cdot c_1\+\cdots \+{\mathbf{\omega }}^{e_{k-1}}\cdot c_{k-1}\+{\mathbf{\omega }}^{e_k}\cdot c_k$, then the full factored normal form is:

where $d_k=e_k$ and $e_{i+1}=e_i+d_i$ for $1\le i<k$.

Each factor $b_i\&equals;{\mathbf{\omega }}^{d_i}\&plus;1$ is irreducible in the sense that if $b_i\&equals;u\cdot v$ for some ordinals $u$ and $v$, then necessarily $u=1$ or $v=1$, and if $b_i=u^v$ for some ordinals $u$ and $v$, then necessarily $u=b_i$ and $v=1$.

The monic factored normal form is:

The rmonic factored normal form is:

If form=pairs is specified, then the result is returned in the form $\left[\left[d_k\&comma;c_k\right]\&comma;\left[d_{k-1}\&comma;c_{k-1}\right]\&comma;\mathrm{...}\&comma;\left[d_1\&comma;c_1\right]\right]$.

The ordinal $a$ can be parametric. However, unless all coefficients $c_i$ are positive when substituting arbitrary nonnegative integers for all the parameters, an error will be raised.

$\mathrm{with}\left(\mathrm{Ordinals}\right)$

$\left[\mathrm{`+`}\&comma;\mathrm{`.`}\&comma;\mathrm{`<`}\&comma;\mathrm{<=}\&comma;\mathrm{Add}\&comma;\mathrm{Base}\&comma;\mathrm{Dec}\&comma;\mathrm{Decompose}\&comma;\mathrm{Div}\&comma;\mathrm{Eval}\&comma;\mathrm{Factor}\&comma;\mathrm{Gcd}\&comma;\mathrm{Lcm}\&comma;\mathrm{LessThan}\&comma;\mathrm{Log}\&comma;\mathrm{Max}\&comma;\mathrm{Min}\&comma;\mathrm{Mult}\&comma;\mathrm{Ordinal}\&comma;\mathrm{Power}\&comma;\mathrm{Split}\&comma;\mathrm{Sub}\&comma;\mathrm{`^`}\&comma;\mathrm{degree}\&comma;\mathrm{lcoeff}\&comma;\log \&comma;\mathrm{lterm}\&comma;\mathrm{\omega }\&comma;\mathrm{quo}\&comma;\mathrm{rem}\&comma;\mathrm{tcoeff}\&comma;\mathrm{tdegree}\&comma;\mathrm{tterm}\right]$

$a≔\mathrm{Ordinal}\left(\left[\left[\mathrm{\omega }\&comma;5\right]\&comma;\left[9\&comma;4\right]\&comma;\left[7\&comma;3\right]\&comma;\left[5\&comma;3\right]\&comma;\left[3\&comma;3\right]\&comma;\left[2\&comma;2\right]\right]\right)$

$a≔{\mathbf{\omega }}^{\mathbf{\omega }}\cdot 5\&plus;{\mathbf{\omega }}^9\cdot 4\&plus;{\mathbf{\omega }}^7\cdot 3\&plus;{\mathbf{\omega }}^5\cdot 3\&plus;{\mathbf{\omega }}^3\cdot 3\&plus;{\mathbf{\omega }}^2\cdot 2$

$\mathrm{Factor}\left(a\right)$

$\left[{\mathbf{\omega }}^2\&comma;2\&comma;\mathbf{\omega }\&plus;1\&comma;3\&comma;{\mathbf{\omega }}^2\&plus;1\&comma;3\&comma;{\mathbf{\omega }}^2\&plus;1\&comma;3\&comma;{\mathbf{\omega }}^2\&plus;1\&comma;4\&comma;{\mathbf{\omega }}^{\mathbf{\omega }}\&plus;1\&comma;5\right]$

$\mathrm{Factor}\left(a\&comma;\mathrm{output}=\mathrm{inert}\right)$

${\mathbf{\omega }}^2\mathbf{\cdot }2\mathbf{\cdot }\left(\mathbf{\omega }\&plus;1\right)\mathbf{\cdot }3\mathbf{\cdot }\left({\mathbf{\omega }}^2\&plus;1\right)\mathbf{\cdot }3\mathbf{\cdot }\left({\mathbf{\omega }}^2\&plus;1\right)\mathbf{\cdot }3\mathbf{\cdot }\left({\mathbf{\omega }}^2\&plus;1\right)\mathbf{\cdot }4\mathbf{\cdot }\left({\mathbf{\omega }}^{\mathbf{\omega }}\&plus;1\right)\mathbf{\cdot }5$

$\mathrm{value}\left(\right)$

${\mathbf{\omega }}^{\mathbf{\omega }}\cdot 5\&plus;{\mathbf{\omega }}^9\cdot 4\&plus;{\mathbf{\omega }}^7\cdot 3\&plus;{\mathbf{\omega }}^5\cdot 3\&plus;{\mathbf{\omega }}^3\cdot 3\&plus;{\mathbf{\omega }}^2\cdot 2$

$\mathrm{Factor}\left(a\&comma;\mathrm{output}=\mathrm{inert}\&comma;\mathrm{form}=\mathrm{monic}\right)$

${\mathbf{\omega }}^2\mathbf{\cdot }\left(\mathbf{\omega }\&plus;2\right)\mathbf{\cdot }{\left({\mathbf{\omega }}^2\&plus;3\right)}^3\mathbf{\cdot }\left({\mathbf{\omega }}^{\mathbf{\omega }}\&plus;4\right)\mathbf{\cdot }5$

$\mathrm{Factor}\left(a\&comma;\mathrm{output}=\mathrm{inert}\&comma;\mathrm{form}=\mathrm{rmonic}\right)$

$\left({\mathbf{\omega }}^2\cdot 2\right)\mathbf{\cdot }\left(\mathbf{\omega }\cdot 3\&plus;1\right)\mathbf{\cdot }{\left({\mathbf{\omega }}^2\cdot 3\&plus;1\right)}^2\mathbf{\cdot }\left({\mathbf{\omega }}^2\cdot 4\&plus;1\right)\mathbf{\cdot }\left({\mathbf{\omega }}^{\mathbf{\omega }}\cdot 5\&plus;1\right)$

$\mathrm{Factor}\left(a\&comma;\mathrm{form}=\mathrm{pairs}\right)$

$\left[\left[2\&comma;2\right]\&comma;\left[1\&comma;3\right]\&comma;\left[2\&comma;3\right]\&comma;\left[2\&comma;3\right]\&comma;\left[2\&comma;4\right]\&comma;\left[\mathbf{\omega }\&comma;5\right]\right]$

$\mathrm{op}\left(a\right)$

$\left[\left[\mathbf{\omega }\&comma;5\right]\&comma;\left[9\&comma;4\right]\&comma;\left[7\&comma;3\right]\&comma;\left[5\&comma;3\right]\&comma;\left[3\&comma;3\right]\&comma;\left[2\&comma;2\right]\right]$

$\mathrm{Factor}\left(a+x\right)$

Error, (in Ordinals:-Factor) cannot determine if x is nonzero

$\mathrm{Factor}\left(a+x+7\&comma;\mathrm{form}=\mathrm{rmonic}\right)$

$\left[x+7\&comma;{\mathbf{\omega }}^2\cdot 2\&plus;1\&comma;\mathbf{\omega }\cdot 3\&plus;1\&comma;{\mathbf{\omega }}^2\cdot 3\&plus;1\&comma;{\mathbf{\omega }}^2\cdot 3\&plus;1\&comma;{\mathbf{\omega }}^2\cdot 4\&plus;1\&comma;{\mathbf{\omega }}^{\mathbf{\omega }}\cdot 5\&plus;1\right]$

$\mathrm{Mult}\left(\mathrm{op}\left(\right)\right)$

${\mathbf{\omega }}^{\mathbf{\omega }}\cdot 5\&plus;{\mathbf{\omega }}^9\cdot 4\&plus;{\mathbf{\omega }}^7\cdot 3\&plus;{\mathbf{\omega }}^5\cdot 3\&plus;{\mathbf{\omega }}^3\cdot 3\&plus;{\mathbf{\omega }}^2\cdot 2\&plus;\left(x+7\right)$

The Ordinals[Factor] command was introduced in Maple 2015.

For more information on Maple 2015 changes, see Updates in Maple 2015.