inert pseudo-remainder function
inert sparse pseudo-remainder function
Prem(a, b, x, 'm', 'q')
Sprem(a, b, x, 'm', 'q')
multivariate polynomials in the variable x
(optional) unevaluated names
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 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. 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.
Prem uses a power of the leading coefficient to the degree difference for the multiplier
Sprem uses a smaller power of the leading coefficient for the multiplier
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