Homotopies for solving problems in enumerative geometry appeared in [44]. The algorithms in the numerical Schubert calculus originated from questions in real enumerative geometry [96,97] and have their main application to the pole placement problem [12], [83,84], [88], [90] in control theory.
The enumerative problems are formalized in some ``finiteness'' theorems, avoiding the explicit but involved (as in (6)) formulas for the root counts.
The first homotopy presented in [44] uses a Gröbner basis for the ideal that defines Gmr, as is derived in [101]. By Gröbner bases questions concerning any polynomial system are solved by relation to monomial equations. Every Gröbner basis defines a flat deformation, which preserves the structure of the solution set [22]. Geometrically, this type of deformation is used to collapse the solution set in projective space to the coordinate hyperplanes, or in the opposite direction, to extend the solutions of the monomial equations to those of the original system. The flat deformations that are obtained in this way are similar to toric deformations in the sense that one moves from the solutions of a subsystem to the solutions of the whole system.
The Gröbner homotopies of [44] work in the synthetic Plücker embedding, and need to take the large set of defining equations of Gmr into account. When expanding the minors into local coordinates, these equations are automatically satisfied, which leads to a much smaller polynomial system. Consequently, the second type of homotopies of [44], the so-called SAGBI homotopies are more efficient. Instead of an ideal, we now have a subalgebra and work with a SAGBI basis, i.e.: the Subalgebra Analogue to a Gröbner Basis for an Ideal. The term order selects the monomials on the diagonal as the dominant ones. This implies that in the flat deformation (see [103] for a general description) only the diagonal monomials remain at t=0.
For m=2=r, the equations of the SAGBI homotopy in determinantal form are
SAGBI homotopies have been implemented [114] to verify some large instances of input planes for which it was conjectured that all solution planes would be real. We refer to [89] and [98] for related work on these conjectures. In [99] an asymptotic choice of inputs is generated for which all solutions are proven to be real.
The third type of homotopies presented in [44] are the so-called Pieri homotopies. Since they are closer to the intrinsic geometric nature, they are applicable to a broader class of problems. In particular, we obtain an effective proof for the following.
Figure 11 shows a part of a cellular decomposition of G22 with the determinantal equations. We specialize the pattern X that represents a solution line by setting some of its coordinates to zero. This specialization determines a specialization of the input lines as follows: take those basis vectors not indexed by rows of Xwhere zeroes have been introduced. The special line SX for this example is as in (31) spanned by the first and third standard basis vector.
Figure 11 pictures patterns of the moving 2-planes in the Pieri homotopy algorithm for the case (m,r) = (2,2), see Figure 6. The bottom matrix is the general representation of a solution that intersects already the two input lines spanned by the standard basis vectors. At the leaves of the tree by linear algebra operations we can intersect with a third input line. Moving down the poset, we deform form the left configuration in Figure 6 to the general problem.
Denote by L1 and L2 the lines already met by X.
At the leaves of the tree in Figure 11
we intersect with the fourth line L4.
The special position for the third line L3 is represented by the
matrix SX, which intersects any X with coordinates as at the leaves
of the tree.
In the homotopy
we deform the line spanned by the columns
of SX to line L3, for t = 0 to t = 1.
The homotopy (31) deforms two solution lines, starting at patterns which have their row indices for the lowest nonzero entries respectively as in [1 4] and in [2 3]. The correctness of this homotopy (proven in [44] and [46]) justifies the formal root count using the localization poset. This combinatorial root count proceeds in two stages. First we build up the poset, from the bottom up, diminishing the entries in the brackets under the restriction that the same entry never occurs twice or more. Secondly, we descend from the top of the poset, collecting and adding up the counts at the nodes in the poset. More examples and variations are in [46].
To solve the general intersection problem of Theorem 5.6, the special (m+1-ki)-planes lie in the intersection of special m-planes. In the construction of the poset one has to follow additional rules as to ensure a solution that meets the intersection of special m-planes. We refer to [46] for details.
The third enumerative problem we can solve is formalized as follows.
The analogue to Figure 11 for maps of degree one into G22 is displayed in Figure 12.
To solve the problem in Theorem 5.7 we need a special position for the interpolation points. By moving those to infinity, the dominant monomials in the maps allow to re-use the same special m-planes, whose entries should be considered modulo m+r. The homotopy to satisfy the n-th intersection condition is:
See [100] for information on the selection of the input planes so that all maps are real.
As an example of another problem in enumerative geometry we mention the 27 lines on a cubic surface in 3-space. According to [81], this is one of the gems hidden in the rag-bag of projective geometry. In [92], the 27 lines are determined by breaking up the cubic surface into three planes in a continuous way such that each intermediate position is nonsingular. It is shown that this continuous variation is also valid in the real field.