2.ix. Why these equations are interesting
The polynomial systems involved in Shapiro's conjecture are interesting not
only because they arise in both geometry and the control of linear systems,
and are conjectured to have all solutions real, but because they are highly
deficient.
By this we mean that they have far fewer solutions than standard
combinatorial bounds.
We illustrate this in the table below (Verschelde has a similar, but more
intensive, table).
Below each pair m,p, we give the bound on the number of solutions
dm,p, and then two standard combinatorial bounds.
The first is the volume of the Newton Polytope of the polynomials in the
polynomial systems.
This was computed using Jan Verschelde's program
PHC.
The last is the Bézout bound, the degree min(m,p) of the
system raised to the power mp-2, which is the number of equations.
Combinatorial Bounds
vs. dm,p
| m,p
| 2,2 | 3,2 | 4,2 | 5,2 | 6,2 | 7,2 | 8,2 | 9,2
|
|---|
| dm,p
| 2 | 5 | 14
| 42 | 132 | 429
| 1430 | 4862
|
|---|
| Newton Polytope
| 2 | 5 | 18
| 67 | 248 | 919
| 3246 | 12863
|
|---|
| Bézout Bound
| 4 | 16 | 64
| 256 | 1024
| 4096 | 16384 | 65536
|
|---|
| m,p
| 2,3 | 3,3 | 4,3 | 5,3 | 2,4 | 3,4 | 2,5 | 2,6
|
|---|
| dm,p
| 5 | 42
| 462 | 6006
| 14 | 462
| 42
| 132
|
|---|
| Newton Polytope
| 5 | 130
| 3004 | 74645
| 42 | 7156
| 364 | 4136
|
|---|
| Bézout Bound
| 81 | 2187
| 59049 | 1594323
| 4096 | 1048576
| 390625 | 60465776
|
|---|
The
PHC
input files (all named Vmp) and output files (named Omp) used for this table
may be obtained by browsing.
There, the files pX.maple were used to generate the files Vmp.