Mixed Volume Computation

Consider two polynomials in $\mathbb{C}[x, y]$ with support sets
$\{1, x^2 y, x y^2\}$ and $\{1, x, x^2, x^2 y\}$ respectively and with generic coefficients. Find the number of their common zeroes in $(\mathbb{C}^*)^2$ by a mixed volume computation. (This problem was on the first exercise sheet.)


Solution:

Let $f_1$ be supported on the set $\{1,x^2y,xy^2\}$ and let $f_2$ be supported on the set $1,x,x^2,x^2y$. Let $P=\textrm{Newt}(f_1)$ and $Q = \textrm{Newt}(f_2)$, then we have:
$P=$
MV2-1.jpg
$Q=$
MV2-2.jpg.

Taking the Minkowski sum, we have $P+Q=$
MV2-3.jpg

Caclulating the area of the parallelograms, we have $MV(P,Q) = \left|\det \begin{pmatrix} 2 & 2 \\ 0 & 1 \end{pmatrix} \right|+ \left|\det \begin{pmatrix} 1 & 2 \\ -1 & 1 \end{pmatrix} \right| = 2 + 3 = 5$. Since $f_1$ and $f_2$ have generic coefficients, by Berstein's Theorem, we have that $f_1$ and $f_2$ have $5$ common zeroes.

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