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Unbounded convex polyhedra as polynomial images of Euclidean spaces

机译：无限的凸多面型作为欧几里德空间的多项式图像

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摘要

In a previous work we proved that each n-dimensional convex polyhedron K subset of R-n and its relative interior are regular images of R-n. As the image of a non-constant polynomial map is an unbounded semialgebraic set, it is not possible to substitute regular maps by polynomial maps in the previous statement. In this work we determine constructively all unbounded n-dimensional convex polyhedra K subset of R-n that are polynomial images of R-n. We also analyze for which of them the interior Int(K) is a polynomial image of R-n. A discriminating object is the recession cone (C) over right arrow (K) of K. Namely, K is a polynomial image of R-n if and only if (C) over right arrow (K) has dimension n. In addition, Int(K) is a polynomial image of R-n if and only if (C) over right arrow (K) has dimension n and K has no bounded faces of dimension n-1. A key result is an improvement of Pecker's elimination of inequalities to represent semialgebraic sets as projections of algebraic sets. Empirical approaches suggest us that there are "few" polynomial maps that have a concrete convex polyhedron as a polynomial image and that there are even fewer for which it is affordable to show that their images actually correspond to our given convex polyhedron. This search of a "needle in the haystack" justifies somehow the technicalities involved in our constructive proofs.
机译：在先前的工作中，我们证明了R-N的每个N维凸多面体K子集及其相对内部是R-N的常规图像。由于非恒定多项式地图的图像是无界面的半峰集合，因此不可能通过先前语句中的多项式映射替换常规映射。在这项工作中，我们建设性地确定所有无界的N维凸多面型R-N的子集，这是R-N的多项式图像。我们还分析了内部int（k）中的哪一个是R-N的多项式图像。鉴别对象是右箭头（k）的衰减锥（c）。即，k是R-n的多项式图像，如果（c）右箭头（k）具有尺寸n。另外，int（k）是R-N的多项式图像，如果右箭头（k）上的（c）仅具有尺寸n，并且k没有尺寸N-1的有界面。一个关键的结果是改善啄木鸟消除不平等，以代表半衰期集合作为代数集的投影。经验方法表明，有“很少”的多项式地图具有混凝土凸多面体作为多项式图像，并且甚至可以较少，其实际上可以证明它们的图像实际上对应于我们给定的凸多面型。此次搜索“大海捞针”的“针”证明了我们有关我们建设性证明的技术性。

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  • 来源
    《Nature reviews Cancer》 |2019年第2期|共57页
  • 作者

    Fernando Jose F.; Manuel Gamboa Jose; Ueno Carlos;

  • 作者单位

    Univ Complutense Madrid Fac Ciencias Matemat Dept Algebra Geometria &

    Topol E-28040 Madrid Spain;

    Univ Complutense Madrid Fac Ciencias Matemat Dept Algebra Geometria &

    Topol E-28040 Madrid Spain;

    Univ Pisa Dipartimento Matemat Largo Bruno Pontecorvo 5 I-56127 Pisa Italy;

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  • 正文语种 eng
  • 中图分类 肿瘤学;
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