The dual of the polynomial ring

The dual polynomials are represented as an ordered list of monomials, which define the inverse system associated to the polynomial ring. The corresponding type is Polynomial MonomialDualRing C, where C is the type of coefficients. Here we describe the main functionnalities available for these polynomials.

Mmx] 

use "realroot"

1.Ring constructors

Mmx] 

R := ZZ['x,'y]

Mmx] 

D := dual R

Mmx] 

Dual polynomial constructors

Construction of dual elements from strings:

Mmx] 

l1 := D<<"215*x^4+10*x-3232231"

Mmx] 

l2 := polynomial(D,"3*x^3*z-x^2*y+2")

Mmx] 

Arithmetic operations

Arithmetic operations inherited from the coefficient ring are available:

Mmx] 

l1+33455552

Mmx] 

l1+=l2

Mmx] 

l1*l2

Mmx] 

Functions

Coefficients with respect to variable :

Mmx] 

coefficients(l1,0)

Dead

Mmx] 

Coefficients of all the monomials:

Mmx] 

coefficients(l1)

Differentiation with respect to variable

Mmx] 

External product with polynomials

Mmx] 

p := R<<"x^2-x*z+y^3-1"

Mmx] 

l1*p

Mmx] 

p*l1

Mmx] 

Mmx]