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prem

pseudo-remainder of polynomials

sprem

sparse pseudo-remainder of polynomials

 

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 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=lcoeffb&comma;xdegreea&comma;xdegreeb&comma;x+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 smaller, in general, equal to lcoeffb&comma;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.

Examples

ax4&plus;1&colon;bcx2&plus;1&colon;

rprema&comma;b&comma;x&comma;&apos;m&apos;&comma;&apos;q&apos;&colon;

r&comma;m&comma;q

cc2+1,c3,ccx21

(1)

rsprema&comma;b&comma;x&comma;&apos;m&apos;&comma;&apos;q&apos;&colon;

r&comma;m&comma;q

c2+1,c2,cx21

(2)

f4x2&plus;2x&plus;1&colon;g2x&plus;1&colon;

rpremf&comma;g&comma;x&comma;&apos;m&apos;&comma;&apos;q&apos;&colon;

r&comma;m&comma;q

4,4,8x

(3)

rspremf&comma;g&comma;x&comma;&apos;m&apos;&comma;&apos;q&apos;&colon;

r&comma;m&comma;q

1,1,2x

(4)

See Also

Prem

quo

rem

Sprem