从Gale对偶多项式系统获得新的稀少多项式上界
New fewnomial upper bounds from Gale dual polynomial systems
AI总结：
本文通过Gale对偶多项式系统的方法，推导出稀少多项式系统的解数上界，并研究了超曲面的连通分支数。
摘要：
我们证明，一个由n个变量的n个多项式组成的稀少多项式系统，其非退化的正解数量少于(e²+3)2^((k choose 2))n^k/4。这比Khovanskii的稀少多项式界2^((n+k choose 2))(n+1)^(n+k)要小得多。我们通过将原系统减少为k个方程在k个变量上的系统，该系统依赖于原系统中单项式的指数的Gale对偶向量配置。然后我们对这个Gale系统进行解数的上界估计。我们采用这些方法证明，一个由n+k+1个单项式定义的R^n正象限超曲面，其至多有C(k)n^(k-1)个紧连通分支。我们的结果适用于具有实数指数的多项式。
英文摘要：
We show that there are fewer than (e^2+3) 2^(k choose 2) n^k/4 non-degenerate positive solutions to a fewnomial system consisting of n polynomials in n variables having a total of n+k+1 distinct monomials. This is significantly smaller than Khovanskii's fewnomial bound of 2^(n+k choose 2)(n+1)^(n+k). We reduce the original system to a system of k equations in k variables which depends upon the vector configuration Gale dual to the exponents of the monomials in the original system. We then bound the number of solutions to this Gale system. We adapt these methods to show that a hypersurface in the positive orthant of R^n defined by a polynomial with n+k+1 monomials has at most C(k)n^(k-1) compact connected components. Our results hold for polynomials with real exponents.
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