The Newton Polytope
نویسنده
چکیده
This describes the map χ in the exact sequence. The map L in the sequence gives us a matrix such that ImL = kerχ. This matrix will generate a integer lattice Λ ⊆ Zm−n. Such an exact sequence gives rise to many different interpretations. The most familiar interpretation is that of a polytope in R which introduces geometry to the system. We can also view the exact sequence as defining a system of differential equations. The volume of the polytope will be related to the number of solutions to the differential equations. Since the columns of the matrix χ generate a integer lattice, it will be useful to us to introduce a volume form which makes the volume of fundamental regions integral values. For any integer lattice Λ ⊂ Z we consider all the n dimensional simplices with vertices in Λ, an elementary simplex will be an n simplex in Λ with minimal volume. The volume form on Λ will give elementary simplices volume 1. Equivalently, this volume form give
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