# Polynomial partitioning for several sets of varieties

@article{Blagojevic2016PolynomialPF, title={Polynomial partitioning for several sets of varieties}, author={Pavle V. M. Blagojevic and Aleksandra Dimitrijevic Blagojevic and G{\"u}nter M. Ziegler}, journal={Journal of Fixed Point Theory and Applications}, year={2016}, volume={19}, pages={1653-1660} }

We give a new, systematic proof for a recent result of Larry Guth and thus also extend the result to a setting with several families of varieties: For any integer $$D\ge 1$$D≥1 and any collection of sets $$\Gamma _1,\ldots ,\Gamma _j$$Γ1,…,Γj of low-degree k-dimensional varieties in $$\mathbb {R}^n$$Rn, there exists a non-zero polynomial $$p\in \mathbb {R}[X_1,\ldots ,X_n]$$p∈R[X1,…,Xn] of degree at most D, so that each connected component of $$\mathbb {R}^n{\setminus }Z(p)$$Rn\Z(p) intersects…

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