3.v. Proof in some cases

   We also have a number of specific cases proven, which we summarize in the following table. For this, we introduce the following notation: Given a Schubert condition a and a positive integer j, (a)^j means that a is repeated j times. Also, set J_i to be the sequence
                    J_i  =   1, 2, ..., p-1, p+i .
The Schubert condition J_i is a Pieri-type condition, it is the condition that a general p-plane H meet a fixed m+1-i-plane nontrivially.

Cases Proven:

Schubert data	(J2)4	(J2)4, J1	(1,3,5)2, (J1)3	(1,3,5)4
m,p	4,2	3,3	3,3	4,3
d	3	3	6	8

   These are all proven in the same way as the proof of Conjecture 1 when (m,p)=(2,3). That is, we consider an equivalent system of polynomials (as described in Section 3.iii) in the local coordinates X_a,b. From this, we compute a universal eliminant and then show that its discriminant is a sum of squares. The links in this table to the data m,p are to documented Maple V.5 scripts detailing these calculations, and the links to the number d are to outputs of these scripts.

   The case (J₂)⁴, J₁ is interesting in that the discriminant is not symmetric in the 3 parameters. The case (1,3,5)², (J₁)³ is the first we showed where the conditions were not just `Pieri-type'.

   The last case of (1,3,5)⁴ with (m,p) = (4,3) is special. Maple was completely unable to handle this on its own. The Maple script in the link generates a Singular input file, given here. This computes a Gröbner basis, but not an eliminant, as that was also too hard. This Gröbner basis is stored as a Maple V.5 file, and we have a Maple V.5 script that reads these equations, and (cleverly) computes the universal eliminant. This is the first case in which the universal eliminant factored---which suggests that the Galois group of this enumerative problem is not the full symmetric group on 8 elements.