By H. Jerome Keisler
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Extra resources for An Infinitesimal Approach to Stochastic Analysis
Example text
Proof. The resultant Resx (f, g) is an irreducible polynomial of degree 2d in a0 , . . , ad , b0 , . . , bd . The determinant of C is also a polynomial of degree 2d. We will show that the zero set of Resx (f, g) is contained in the zero set of det(C). This implies that the two polynomials are equal up to a constant. By examining the leading terms of both polynomials in the lexicographic term order, we find that the constant is either 1 or −1. If (a0 , . . , ad , b0 , . . , bd ) is in the zero set of Resx (f, g) then the system f = g = 0 has a complex solution x0 .
For instance, if we have three polytopes in R3 , each of which is the sum of two other polytopes, then multilinearity says: M(P1 + Q1 , P2 + Q2 , P3 + Q3 ) = M(P1 , P2 , P3 ) + M(P1 , P2 , Q3 ) + M(P1 , Q2 , P3 ) + M(P1 , Q2 , Q3 ) + M(Q1 , P2 , P3 ) + M(Q1 , P2 , Q3 ) + M(Q1 , Q2 , P3 ) + M(Q1 , Q2 , Q3 ). This sum of eight smaller mixed volumes reflects the fact that the number of roots of system of equations is additive when each equation factors as in f1 (x, y, z) · g1 (x, y, z) f2 (x, y, z) · g2 (x, y, z) f3 (x, y, z) · g3 (x, y, z) = = = 0, 0, 0.
3 of [Sha94]. We now choose c by choosing (f1 , . . , fn ) as follows. Let f1 , . . 4) which have only finitely many zeros in Pn−1 . Then choose fn which vanishes at exactly one of these zeros, say y ∈ Pn−1 . Hence ψ −1 (c) = {(c, y)}, a zero-dimensional variety. For this particular choice of c both inequalities hold with equality. This implies dim(ψ(I)) = N − 1. We have shown that the image of I under ψ is an irreducible hypersurface in CN , which is defined over Z. Hence there exists an irreducible polynomial Res ∈ Z[c], unique up to sign, whose zero set equals ψ(I).