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, , the name (such as "a") to be used for the variables of the Lazard ring
• t, , the name (such as "x") of the first variable of the formal group law
• u, , the name (such as "y") of the second variable of the formal group law
• Outputs:

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.