universalFGL(ZZ,String,String,String) -- universal formal group law in the Lazard ring

Synopsis

Function: universalFGL
Usage:

universalFGL(n,s,t,u)
Inputs:
- n, an integer, the degree of precision
- s, a string, the name (such as "a") to be used for the variables of the Lazard ring
- t, a string, the name (such as "x") of the first variable of the formal group law
- u, a string, the name (such as "y") of the second variable of the formal group law
Outputs:
- an instance of the type FormalGroupLaw

Description

The following returns the formal group law over the Lazard ring (seen as a polynomial ring in the {a_i}'s up to degree n.

i1 : universalFGL(3,"a","x","y")

                       2         2
o1 = FormalGroupLaw{a x y + a x*y  + a x*y + x + y, 3}
                     2       2        1

o1 : FormalGroupLaw

i2 : universalFGL(4,"a","x","y")

                                    3                    2 2                     3      2         2
o2 = FormalGroupLaw{(- 2a a  + 2a )x y + (- 2a a  + 3a )x y  + (- 2a a  + 2a )x*y  + a x y + a x*y  + a x*y + x + y, 4}
                         1 2     3            1 2     3             1 2     3         2       2        1

o2 : FormalGroupLaw

Caveat

The decomposition of the Lazard as a polynomial ring in an infinite number of variables is not canonical, we have made a choice, here, which amounts to choosing, for every d at most n, of Bezout coefficients for the set of binomial coefficients (d,i), 1<i<d. Variables with names equal to the strings (like x, y or a, here) should not have been assigned values (like 3) beforehand otherwise an error will occur.