Given a polynomial p expressed as a Chebyshev series, determine the degree of the polynomial (i.e. the largest k such that $T\left(k\,x\right)$ appears as a basis polynomial).

All Chebyshev basis polynomials $T\left(k\,x\right)$ which appear must have the same second argument x (which can be any expression).

The input polynomial must be in expanded form (i.e. a sum of products). Normally, each term in the sum contains one and only one $T\left(k\,x\right)$ factor except that if there are terms in the sum containing no $T\left(k\,x\right)$ factor then each such term t is interpreted to represent $tT\left(0\,x\right)$ (i.e. it is assumed to be a term of degree 0).

The command with(numapprox,chebdeg) allows the use of the abbreviated form of this command.

$\mathrm{with}\left(\mathrm{numapprox}\right)\:$

$\mathrm{Digits}≔3\:$

$a≔\mathrm{chebyshev}\left(\sin \left(x\right)\,x\right)\:$

$b≔\mathrm{chebyshev}\left(\cos \left(x\right)\,x\right)\:$

$c≔a+b$

$c≔0.880T\left(1\,x\right)-0.0391T\left(3\,x\right)+0.000500T\left(5\,x\right)+0.765T\left(0\,x\right)-0.230T\left(2\,x\right)+0.00495T\left(4\,x\right)$

$\mathrm{chebdeg}\left(c\right)$

$5$

$d≔a+\mathrm{cj}T\left(j\,x\right)+\mathrm{ck}T\left(k\,x\right)$

$d≔0.880T\left(1\,x\right)-0.0391T\left(3\,x\right)+0.000500T\left(5\,x\right)+\mathrm{cj}T\left(j\,x\right)+\mathrm{ck}T\left(k\,x\right)$

$\mathrm{chebdeg}\left(d\right)$

$\max \left(5\,j\,k\right)$

$\mathrm{assume}\left(5<k\&comma;k<j\right)$

$e≔1.2y+\mathrm{cj}T\left(j\&comma;x\right)+a+\mathrm{ck}T\left(k\&comma;x\right)$

$e≔1.2y+\mathrm{cj}T\left(\mathrm{j~}\&comma;x\right)+0.880T\left(1\&comma;x\right)-0.0391T\left(3\&comma;x\right)+0.000500T\left(5\&comma;x\right)+\mathrm{ck}T\left(\mathrm{k~}\&comma;x\right)$

$\mathrm{chebdeg}\left(e\right)$

$\mathrm{j~}$