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Prem

inert pseudo-remainder function

Sprem

inert sparse pseudo-remainder function

 

Calling Sequence

Parameters

Description

Examples

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 degreer&comma;x<degreeb&comma;x and m (the multiplier) is:

m&equals;lcoeffb&comma;xdegreea&comma;xdegreeb&comma;x&plus;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 lcoeffb&comma;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.

Examples

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

Premx101&comma;yx21&comma;x&comma;&apos;m&apos;mod13

12y9&plus;y4

(1)

m

y9

(2)

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

Spremx101&comma;yx21&comma;x&comma;&apos;m&apos;&comma;&apos;q&apos;mod13

12y5&plus;1

(3)

m

y5

(4)

q

x8y4&plus;x6y3&plus;x4y2&plus;x2y&plus;1

(5)

See Also

evala

mod

prem

Rem

RootOf

sprem

 


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