The Prem and Sprem functions are placeholders for the pseudo-remainder and sparse pseudo-remainder of a divided by b where a and b are polynomials in the variable x. They are used in conjunction with either mod or evala which define the coefficient domain, as described below.

The function Prem returns the pseudo-remainder r such that:

where $\mathrm{degree}\left(r\&comma;x\right)<\mathrm{degree}\left(b\&comma;x\right)$ 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 $\mathrm{lcoeff}\left(b\&comma;x\right)$ to the power of the number of division steps performed rather than the degree difference. When Sprem can be used it is preferred because it is more efficient.

The calls  Prem(a, b, x, 'm', 'q') mod p and Sprem(a, b, x, 'm', 'q') mod p compute the pseudo-remainder and sparse pseudo-remainder respectively of a divided  by b modulo p, a prime integer. The coefficients of a and b must be multivariate polynomials over the rationals or coefficients over a finite field specified by RootOf expressions.

The calls evala(Prem(a, b, x, 'm', 'q')) and evala(Sprem(a, b, x, 'm', 'q')) compute the pseudo-remainder and sparse pseudo-remainder respectively of a and b, where the coefficients of a and b are multivariate polynomials with coefficients in an algebraic number (or function) field.

$\mathrm{Prem}\left(x^{10}-1\&comma;y{x}^2-1\&comma;x\&comma;'m'\right)\mathbf{mod}13$

$12y^9+y^4$

$m$

$y^9$

$\mathrm{Sprem}\left(x^{10}-1\&comma;y{x}^2-1\&comma;x\&comma;'m'\&comma;'q'\right)\mathbf{mod}13$

$12y^5+1$

$m$

$y^5$

$q$

$x^8{y}^4+x^6{y}^3+x^4{y}^2+y{x}^2+1$