We initiate a study of the Euclidean distance degree in the context of sparse polynomials. Specifically, we consider a hypersurface \(f=0\) defined by a polynomial f that is general given its support, such that the support contains the origin. We show that the Euclidean distance degree of \(f=0\) equals the mixed volume of the Newton polytopes of the associated Lagrange multiplier equations. We discuss the implication of our result for computational complexity and give a formula for the Euclidean distance degree when the Newton polytope is a rectangular parallelepiped.

中文翻译：

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欧氏距离度和混合体积

我们在稀疏多项式的背景下开始研究欧几里得距离度。具体来说，我们考虑由多项式f定义的超曲面\(f=0\)，该多项式 f是通用的，其支持包含原点。我们表明\(f=0\)的欧几里得距离度等于相关拉格朗日乘子方程的牛顿多胞体的混合体积。我们讨论了我们的结果对计算复杂性的影响，并给出了当牛顿多胞体是长方体时欧几里得距离度的公式。