If x is a single indeterminate, the degree and ldegree commands compute the degree and low degree, respectively, of the polynomial a in x. If x is not specified then the degree and ldegree commands compute the total degree and total low degree, respectively, of the polynomial a in all of its indeterminates. The definitions for the cases where x is a list or set of indeterminates are given below.

The polynomial a can have negative integer exponents in x. Thus degree and ldegree functions can return a negative or positive integer.  If a is not a polynomial in x in this generalized sense, then FAIL is returned.

The identically 0 polynomial is defined to have degree -infinity and ldegree +infinity.

The polynomial a must be in collected form in order for degree/ldegree to return an accurate result.  For example, given $\left(x+1\right)\left(x+2\right)-x^2$, degree would not detect the cancellation of the leading term, and would incorrectly return a result of 2.  Applying collect with normalization or expand to the polynomial before calling degree avoids this problem.

If x is a set of indeterminates, the total degree/ldegree is computed.  If x is a list of indeterminates, then the vector degree/ldegree is computed. Finally, if x is not specified, this is short for degree(a,indets(a)), meaning that the total degree in all the indeterminates is computed. The vector degree is defined as follows:

The total degree is then defined as

Notice that the vector degree is sensitive to the order of the indeterminates, whereas the total degree is not.

$a≔x^4-10x^2+1$

$a≔x^4-10x^2+1$

$\mathrm{degree}\left(a\,x\right)$

$4$

$\mathrm{ldegree}\left(a\,x\right)$

$0$

$b≔x^{-2}-2+3x$

$b≔\frac{1}{x^2}-2+3x$

$\mathrm{degree}\left(b\,x\right)$

$1$

$\mathrm{ldegree}\left(b\,x\right)$

$\mathrm{-2}$

$c≔x^{2}y+3x{y}^2+x^3{y}^3-x^5$

$c≔x^3{y}^3-x^5+x^{2}y+3x{y}^2$

$\mathrm{degree}\left(c\,x\right)$

$5$

$\mathrm{degree}\left(c\,y\right)$

$3$

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

$6$

$\mathrm{ldegree}\left(c\,x\right)$

$1$

$\mathrm{ldegree}\left(c\,y\right)$

$0$

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

$3$

$f≔x{y}^3+x^2$

$f≔x{y}^3+x^2$

$\mathrm{degree}\left(f\,x\right),\mathrm{degree}\left(f\,y\right)$

$2,3$

$\mathrm{degree}\left(f\right)$

$4$

$\mathrm{degree}\left(f\,\left\{x\,y\right\}\right)$

$4$

$\mathrm{degree}\left(f\,\left\{x\,y\right\}\right)$

$4$

$\mathrm{degree}\left(f\,\left[x\,y\right]\right)$

$2$

$\mathrm{degree}\left(f\,\left[y\,x\right]\right)$

$4$

$\mathrm{degree}\left(y\sin \left(x\right)\,x\right)$

$\mathrm{FAIL}$

$\mathrm{degree}\left(y\sin \left(x\right)\,y\right)$

$1$

$\mathrm{degree}\left(\frac{x+1}{x+2}\,x\right)$

$\mathrm{FAIL}$

$\mathrm{zero}≔y\left(\frac{x}{x+1}+\frac{1}{x+1}-1\right)$

$\mathrm{zero}≔y\left(\frac{x}{x+1}+\frac{1}{x+1}-1\right)$

$\mathrm{degree}\left(\mathrm{zero}\,x\right)$

$\mathrm{FAIL}$

$\mathrm{degree}\left(\mathrm{zero}\,y\right)$

$1$

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

$0$

$\mathrm{degree}\left(\mathrm{collect}\left(\mathrm{zero}\,x\,\mathrm{normal}\right)\,x\right)$

$-\mathrm{\infty }$

$\mathrm{degree}\left(\mathrm{collect}\left(\mathrm{zero}\,y\,\mathrm{normal}\right)\,y\right)$

$-\mathrm{\infty }$