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A $k$-ary semialgebraic predicate $\\Phi(x_1,\\ldots,x_k)$ on $\\mathbb{R}^d$ is a Boolean combination of polynomial equations and inequalities in the $kd$ coordinates of $k$ points $x_1,\\ldots,x_k\\in\\mathbb{R}^d$. A sequence $P=(p_1,\\ldots,p_n)$ of points in $\\mathbb{R}^d$ is called $\\Phi$-homogeneous if either $\\Phi(p_{i_1}, \\ldots,p_{i_k})$ holds for all choices $1\\le i_1 < \\cdots < i_k\\le n$, or it holds for no such choice. 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