> <\body> <\tmdoc-title> The dual of the polynomial ring The dual polynomials are represented as an ordered list of monomials, which define the associated to the polynomial ring. The corresponding type is , where is the type of coefficients. Here we describe the main functionnalities available for these polynomials.\ <\session|mathemagix|default> <\input> <|input> use "realroot" <\session|mathemagix|default> <\unfolded-io> <|unfolded-io> R := ZZ['x,'y] <|unfolded-io> > <\unfolded-io> <|unfolded-io> D := dual R <|unfolded-io> >> <\input> <|input> \; \; <\unfolded> <|unfolded> <\textput> Construction of dual elements from strings: <\unfolded-io> <|unfolded-io> l1 := D\\"215*x^4+10*x-3232231"\ <|unfolded-io> > <\unfolded-io> <|unfolded-io> l2 := polynomial(D,"3*x^3*z-x^2*y+2") <|unfolded-io> *dy+3*dx*dz> <\input> <|input> \; \; <\unfolded> <|unfolded> <\textput> Arithmetic operations inherited from the coefficient ring are available: <\unfolded-io> <|unfolded-io> l1+33455552 <|unfolded-io> > <\unfolded-io> <|unfolded-io> l1+=l2 <|unfolded-io> *dy+3*dx*dz+215*dx> <\unfolded-io> <|unfolded-io> l1*l2 <|unfolded-io> *dy-9696681*dx*dz-10*dx*dy+430*dx+30*dx*dz+dx*dy-6*dx*dy*dz-215*dx*dy+9*dx*dz+645*dx*dz> <\input> <|input> \; \; <\unfolded> <|unfolded> <\textput> Coefficients with respect to variable >: <\unfolded-io> <|unfolded-io> coefficients(l1,0) <|unfolded-io> <\input|> \; \; <\textput> <\textput> Coefficients of all the monomials: <\unfolded-io> <|unfolded-io> coefficients(l1) <|unfolded-io> > \; <\textput> <\textput> Differentiation with respect to variable > <\input> <|input> \; \; <\unfolded> <|unfolded> \; <\unfolded-io> <|unfolded-io> p := R\\"x^2-x*z+y^3-1" <|unfolded-io> +x-x*z-1> <\unfolded-io> <|unfolded-io> l1*p <|unfolded-io> -3232229*x+3232229*x*z+10*x-10*z+3232229> <\unfolded-io> <|unfolded-io> p*l1 <|unfolded-io> +dx*dy-3*dx*dz-215*dx> <\input> <|input> \; \; <\input|> \; \; <\initial> <\collection>