Decomposing Solution Sets of Polynomial Systems Using Derivatives

Trace Test

“…Suppose that V ⊂ C n 1 × C n 2 is an irreducible variety of dimension m > 0. Letting z (i) be coordinates for C n i for i = 1, 2, the variety V is defined by polynomials F (z (1) , z (2) ) which generate a prime ideal. Separately homogenizing these polynomials in each set z (i) of variables gives bihomogeneous polynomials that define the closure V of V in the product P n 1 × P n 2 of projective spaces.…”

“…This is a set of nonnegative integers d m 1 ,m 2 where m 1 + m 2 = m with 0 ≤ m i ≤ n i for i = 1, 2 that has the following geometric meaning. Given general linear subspaces M i ⊂ P n i of codimension m i for i = 1, 2 with m 1 + m 2 = m, the number of points in the intersection V ∩ (M (1) × M (2) ) is d m 1 ,m 2 . Multidegrees are log-concave in that for every 1 ≤ m 1 ≤ m−1, we have…”

“…, where for i = 1, 2, M (i) ⊂ P n i is a general linear subspace of codimension m i . More formally, the witness set is a triple consisting of the points W m 1 ,m 2 , equations for a variety that has V as a component, and equations for M (1) and for M (2) . A witness collection is the list of witness sets W m 1 ,m 2 for all m 1 + m 2 = m.…”

“…Methods to check linearity of a univariate function-e.g., the trace-in the context of algorithms for numerical algebraic geometry were recently discussed in [2].…”

“…Witness sets for multihomogeneous varieties were introduced in [4]. Figure 3 shows that the trace obtained by varying ℓ (2) is nonlinear in either affine chart. One may instead apply the trace test to a witness set in the ambient projective space of the Segre embedding.…”