prem - Maple Help

prem

pseudo-remainder of polynomials

sprem

sparse pseudo-remainder of polynomials

 Calling Sequence prem(a, b, x, 'm', 'q') sprem(a, b, x, 'm', 'q')

Parameters

 a, b - multivariate polynomials in the variable x x - indeterminate m, q - (optional) unevaluated names

Description

 • The function prem  returns the pseudo-remainder r such that

$ma=bq+r$

 where $\mathrm{degree}\left(r,x\right)<\mathrm{degree}\left(b,x\right)$ and m (the multiplier) is:

$m={\mathrm{lcoeff}\left(b,x\right)}^{\mathrm{degree}\left(a,x\right)-\mathrm{degree}\left(b,x\right)+1}$

 • If the fourth argument is present it is assigned the value of the multiplier m defined above.  If the fifth argument is present, it is assigned the pseudo-quotient q defined above.
 • The function sprem has the same functionality as prem except that the multiplier m will be smaller, in general, equal to $\mathrm{lcoeff}\left(b,x\right)$ to the power of the number of division steps performed rather than the degree difference. If both $a$ and $b$ are multivariate polynomials with integer coefficients, then m is the (unique) smallest possible multiplier with positive leading coefficient that makes the pseudo-division fraction free.
 • When sprem can be used it is preferred over prem because it is more efficient.

Examples

 > $a≔{x}^{4}+1:$$b≔c{x}^{2}+1:$
 > $r≔\mathrm{prem}\left(a,b,x,'m','q'\right):$
 > $r,m,q$
 ${c}{}\left({{c}}^{{2}}{+}{1}\right){,}{{c}}^{{3}}{,}{c}{}\left({c}{}{{x}}^{{2}}{-}{1}\right)$ (1)
 > $r≔\mathrm{sprem}\left(a,b,x,'m','q'\right):$
 > $r,m,q$
 ${{c}}^{{2}}{+}{1}{,}{{c}}^{{2}}{,}{c}{}{{x}}^{{2}}{-}{1}$ (2)
 > $f≔4{x}^{2}+2x+1:$$g≔2x+1:$
 > $r≔\mathrm{prem}\left(f,g,x,'m','q'\right):$
 > $r,m,q$
 ${4}{,}{4}{,}{8}{}{x}$ (3)
 > $r≔\mathrm{sprem}\left(f,g,x,'m','q'\right):$
 > $r,m,q$
 ${1}{,}{1}{,}{2}{}{x}$ (4)