MULTIVARIATE POLYNOMIALS IN SAGE

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12 VIVIANE PONS hal-00824143, version 1 - 21 May 2013 ... return q *1/ t* call _ back (v). divided _ difference (i +1) ... return basis (v) Now that we have the function, we will pass it to our algebra to create a new basis: sage : A = AbstractPolynomialRing (QQ) sage : myBasis = A. linear _ basis _on_ vectors ("A"," MySchub " ,"Y", schubert _on_ basis ) sage : pol = myBasis [2 ,1 ,3]; pol Y(2 , 1, 3) sage : pol . expand () x(2 , 1, 3) + x(2 , 2, 2) + x(2 , 3, 1) + x(3 , 1, 2) + x(3 , 2, 1) + x(4 , 1, 1) sage : myBasis (A.an_ element ()) Y(1 , 2, 3) - Y(1 , 3, 2) - Y(2 , 1, 3) + Y(2 , 3, 1) + Y(3 , 1, 2) - Y(3 , 2, 1) + Y(4 , 1, 1) This is a copy of the Schubert basis, and it works the same way as the previous bases we have seen in Section 3.4. Below is an example with a parametrized function: sage : var (’q t ’) (q, t) sage : K. = QQ [] sage : K = K. fraction _ field () sage : A = AbstractPolynomialRing (K) sage : qtSchubertBasis = A. linear _ basis _on_ vectors ("A"," qtSchub " ," YQ",qt_ schubert _on_basis ,((" q",q) ,("t",t)) ) sage : pol = qtSchubertBasis [1 ,2 ,3]; pol YQ (1 , 2, 3) sage : pol . expand () q ^3/ t ^3* x(1 , 2, 3) + q ^3/ t ^3* x(1 , 3, 2) + q ^3/ t ^3* x(2 , 1, 3) + 2*q^3/ t ^3* x(2 , 2, 2) + q ^3/ t ^3* x(2 , 3, 1) + q ^3/ t ^3* x(3 , 1, 2) + q ^3/ t ^3* x(3 , 2, 1) The extra parameters are sent by a tuple of couples (parameter name, parameter value) to the main algebra that will create the new basis. 3.6. Double set of variables. Our program also contains another algebra to work with a double set of variables. sage : D = DoubleAbstractPolynomialRing (QQ); D The abstract ring of multivariate polynomials on x over The abstract ring of multivariate polynomials on y over Rational Field One can see that the double algebra is the algebra of multivariate polynomials in the x variables with the multivariate polynomials in the y variables as coefficients. sage : D.an_ element () y [0]* x[0 , 0, 0] + 2*y [0]* x[1 , 0, 0] + y [0]* x[1 , 2, 3] + 3*y [0]* x [2 , 0, 0]
MULTIVARIABLE POLYNOMIALS IN SAGE 13 One can specify which bases to use in the x variables and which bases to use in the y variables. hal-00824143, version 1 - 21 May 2013 sage : Dx = D sage : Dy = D. base _ ring () sage : Schubx = Dx. schubert _ basis _on_ vectors () sage : Schuby = Dy. schubert _ basis _on_ vectors () sage : pol = Schuby [2 ,1 ,3] * Schubx [1 ,1 ,2] sage : pol (Yy (2 ,1 ,3) )*Yx (1 , 1, 2) The expand function or all direct conversions are done in the x variables. sage : pol . expand () (Yy (2 ,1 ,3) )*x(1 , 1, 2) + (Yy (2 ,1 ,3) )*x(1 , 2, 1) + (Yy (2 ,1 ,3) )*x (2 , 1, 1) sage : mx = Dx. monomial _ basis () sage : mx(pol ) (Yy (2 ,1 ,3) )*x[1 , 1, 2] + (Yy (2 ,1 ,3) )*x[1 , 2, 1] + (Yy (2 ,1 ,3) )*x [2 , 1, 1] But, of course, one can also easily change the basis for the y variables. sage : my = Dy. monomial _ basis () sage : pol . change _ coeffs _ bases (my) (y [2 ,1 ,3]+ y [2 ,2 ,2]+ y [2 ,3 ,1]+ y [3 ,1 ,2]+ y [3 ,2 ,1]+ y [4 ,1 ,1]) *Yx (1 , 1, 2) sage : pol = mx(pol ); pol (Yy (2 ,1 ,3) )*x[1 , 1, 2] + (Yy (2 ,1 ,3) )*x[1 , 2, 1] + (Yy (2 ,1 ,3) )*x [2 , 1, 1] sage : pol . change _ coeffs _ bases (my) (y [2 ,1 ,3]+ y [2 ,2 ,2]+ y [2 ,3 ,1]+ y [3 ,1 ,2]+ y [3 ,2 ,1]+ y [4 ,1 ,1]) *x[1 , 1, 2] + (y [2 ,1 ,3]+ y [2 ,2 ,2]+ y [2 ,3 ,1]+ y [3 ,1 ,2]+ y [3 ,2 ,1]+ y [4 ,1 ,1]) * x[1 , 2, 1] + (y [2 ,1 ,3]+ y [2 ,2 ,2]+ y [2 ,3 ,1]+ y [3 ,1 ,2]+ y [3 ,2 ,1]+ y [4 ,1 ,1]) *x[2 , 1, 1] One can also change the role of variables between the main ones and the coefficients. sage : pol (Yy (2 ,1 ,3) )*x[1 , 1, 2] + (Yy (2 ,1 ,3) )*x[1 , 2, 1] + (Yy (2 ,1 ,3) )*x [2 , 1, 1] sage : pol . swap _ coeffs _ elements () (x [1 ,1 ,2]+ x [1 ,2 ,1]+ x [2 ,1 ,1]) *Yy (2 , 1, 3) So we have seen that we can use our previous bases on a double set of variables. But we also have specific bases that only work with a double set of variables. Let us see the double Schubert polynomials and double Grothendieck polynomials, the way they were defined in Section 2.3. sage : DoubleSchub = D. double _ schubert _ basis _on_ vectors (); DoubleSchub
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