The functions lcoeff and tcoeff return the leading (trailing) coefficient of p with respect to the indeterminate(s) x or the monomial order o.

If neither x nor o is specified, then lcoeff (tcoeff) computes the leading (trailing) coefficient with respect to all the indeterminates of p.

If a third argument t is specified ("call by name"), it is assigned the leading (trailing) monomial of p.

If x is a single indeterminate, and d is the degree (low degree) of p in x, then lcoeff(p, x) (tcoeff(p, x)) is equivalent to coeff(p, x, d). If x is a list or set of indeterminates, lcoeff (tcoeff) computes the leading (trailing) coefficient of p considered as a multivariate polynomial in the variables x, using lexicographic order. More precisely, lcoeff(p, [x1, ..., xn]) is equivalent to lcoeff(...(lcoeff(p, x1), ...), xn) (and similarly for tcoeff).

Other monomial orders can be specified by using the order=o calling sequence. The supported orders are:

plex(x1, ..., xn) - lexicographic order

grlex(x1, ..., xn) - graded lexicographic order

tdeg(x1, ..., xn) - graded reverse lexicographic order

for indeterminates x1, ..., xn. For a description of these orders, see Monomial orders for multivariate polynomials.

Note that p must be collected with respect to the appropriate indeterminates before calling lcoeff or tcoeff. For details, see collect.

When neither x nor o is specified, the order of the indeterminates is given by indets (more specifically,$\mathrm{frontend}\left(\mathrm{indets},\left[p\right],\left[\left\{\mathrm{`*`},\mathrm{`+`},\mathrm{`::`},\mathrm{constant},\mathrm{series},\mathrm{SDMPolynom},\mathrm{undefined}\right\}\right]\right)$ ). In the multivariate case this ordering may be session dependent.

The lcoeff and tcoeff commands are thread-safe as of Maple 15.

For more information on thread safety, see index/threadsafe.

$s≔3v^2{w}^3{x}^4+1$

$s≔3v^2{w}^3{x}^4+1$

$\mathrm{lcoeff}\left(s\right)$

$3$

$\mathrm{tcoeff}\left(s\right)$

$1$

$\mathrm{lcoeff}\left(s\,\left[v\,w\right]\,'t'\right)$

$3x^4$

$t$

$v^2{w}^3$

$p≔x+4xy+5y-7x^2$

$p≔-7x^2+4xy+x+5y$

$\mathrm{lcoeff}\left(p\right)$

$\mathrm{-7}$

$\mathrm{tcoeff}\left(p\right)$

$5$

$\mathrm{lcoeff}\left(p\,x\right)$

$\mathrm{-7}$

$\mathrm{lcoeff}\left(p\,y\right)$

$4x+5$

$\mathrm{tcoeff}\left(p\,x\right)$

$5y$

$\mathrm{tcoeff}\left(p\,y\right)$

$-7x^2+x$

$\mathrm{collect}\left(p\,x\right)$

$-7x^2+\left(4y+1\right)x+5y$

$\mathrm{collect}\left(p\,y\right)$

$\left(4x+5\right)y-7x^2+x$

$\mathrm{coeff}\left(p\,x\,1\right)$

$4y+1$

$f≔4x^3+5x^2{z}^2+2x{y}^{2}z+1$

$f≔5x^2{z}^2+2x{y}^{2}z+4x^3+1$

$\mathrm{lcoeff}\left(f\,\mathrm{order}=\mathrm{plex}\left(x\,y\,z\right)\,'m'\right),m$

$4,x^3$

$\mathrm{lcoeff}\left(f\,\mathrm{order}=\mathrm{grlex}\left(x\,y\,z\right)\,'m'\right),m$

$5,x^2{z}^2$

$\mathrm{lcoeff}\left(f\,\mathrm{order}=\mathrm{tdeg}\left(x\,y\,z\right)\,'m'\right),m$

$2,x{y}^{2}z$