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## 6.ii Lower bounds in the Schubert calculus for flags?

Conjecture 5.12 speculates that a zero-dimensional intersection of Grassmannian Schubert varieties in a flag manifold Flb has only real points, when the Schubert varieties are given by flags osculating the rational normal curve at points whose associated partition is ordered. If the associated partition is not ordered, then we conjectured (and have found experimentally) that there is a selection of osculating flags with all points of intersection real, as well as a selection with not all points real.

The table below shows the results of computing 160,000 instances of the intersection (5.11), recording the number having a given number of real points, and sorting this by the simple Schubert data. The data is a sequence of 4 2's and 4 3's, up to reversal and cyclic permutations, there are 8 such sequences. Observe that the apparent lower bound on the number of real solutions depends on the sequence, which is a measure of the configuration of the conditions. This reinforces the observation made in Section 3.i concerning the subtle dependence of the number of real solutions on the configuration of the conditions. Other calculations we have done reinforce this observation.

 Data Number of Real Solutions 0 2 4 6 8 10 12 22223333 0 0 0 0 0 0 20000 22332233 0 0 136 1533 7045 5261 6025 22233233 0 0 67 3015 6683 4822 5413 22322333 0 0 9 1677 3835 6247 8232 22323323 0 0 303 2090 6014 7690 3903 22232333 0 195 1476 1776 3628 4546 8379 22323233 0 37 1944 4367 6160 4634 2858 23232323 251 929 5740 3168 5420 2828 1662

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