5.1. Viro’s patchworking [04T1]

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5.1. Viro’s patchworking

Let $v:\Delta\cap\mathbb{Z}^{n+1}\to\mathbb{R}$ be any function and $a(z)=\sum\limits_{j\in\Delta\cap\mathbb{Z}^{n+1}}a_{j}z^{j}$ be any polynomial. Following [15] we define the patchworking polynomial for any $t>0$ by

f^{v}_{t}(z)=\sum\limits_{j\in\Delta\cap\mathbb{Z}^{n+1}}a_{j}t^{-v(j)}z^{j},

where $a_{j}\neq 0$ for any $j\in\Delta\cap\mathbb{Z}^{n+1}$ . Note that if $v$ is integer-valued then $f^{v}_{t}$ makes sense also for any $t\in\mathbb{C}^{*}$ .

Remark 5.1.

In [15] the patchworking polynomial was used for construction of real algebraic hypersurfaces with controlled topology. The topology of the zero set of a real patchworking polynomial for $t>>0$ depends only on the function $v$ and on the signs of the coefficients $a_{j}$ .