pseudo-remainder of polynomials
sparse pseudo-remainder of polynomials
prem(a, b, x, 'm', 'q')
sprem(a, b, x, 'm', 'q')
multivariate polynomials in the variable x
(optional) unevaluated names
The function prem returns the pseudo-remainder r such that
where degree⁡r,x<degree⁡b,x and m (the multiplier) is:
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 lcoeff⁡b,x 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.
a ≔ x4+1:b ≔ c⁢x2+1:
r ≔ prem⁡a,b,x,'m','q':
r ≔ sprem⁡a,b,x,'m','q':
f ≔ 4⁢x2+2⁢x+1:g ≔ 2⁢x+1:
r ≔ prem⁡f,g,x,'m','q':
r ≔ sprem⁡f,g,x,'m','q':
Download Help Document