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Why these equations are interesting Previous Next Contents

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 2,23,24,25,26,27,28,29,2
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 2,33,34,35,32,43,42,52,6
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.

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