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Prem

inert pseudo-remainder function

Sprem

inert sparse pseudo-remainder function

 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 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:

$ma=bq+r$

 • 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 $\mathrm{lcoeff}\left(b,x\right)$ 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.

Examples

Prem uses a power of the leading coefficient to the degree difference for the multiplier

 > $\mathrm{Prem}\left({x}^{10}-1,y{x}^{2}-1,x,'m'\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{mod}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}13$
 ${12}{}{{y}}^{{9}}{+}{{y}}^{{4}}$ (1)
 > $m$
 ${{y}}^{{9}}$ (2)

Sprem uses a smaller power of the leading coefficient for the multiplier

 > $\mathrm{Sprem}\left({x}^{10}-1,y{x}^{2}-1,x,'m','q'\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\mathbf{mod}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}13$
 ${12}{}{{y}}^{{5}}{+}{1}$ (3)
 > $m$
 ${{y}}^{{5}}$ (4)
 > $q$
 ${{x}}^{{8}}{}{{y}}^{{4}}{+}{{x}}^{{6}}{}{{y}}^{{3}}{+}{{x}}^{{4}}{}{{y}}^{{2}}{+}{{x}}^{{2}}{}{y}{+}{1}$ (5)