摘要翻译:
利用多项式完全交集的Gale对偶,并利用正解的fewnomial界的证明,得到了具有n+k+1个单项式的n个变量的n个多项式系统的非零实解个数的界(E^4+3)2^(k选择2)n^k/4,其指数向量生成奇数指数的Z^n子群。该界仅以常数因子(E^4+3)/(E^2+3)超过正解的界,并且对于k固定和n大的情况,该界是渐近尖锐的。
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英文标题:
《Bounds on the number of real solutions to polynomial equations》
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作者:
Daniel J. Bates (IMA), Fr\'ed\'eric Bihan (Universit\'e de Savoie),
and Frank Sottile (Texas A&M)
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最新提交年份:
2007
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分类信息:
一级分类:Mathematics 数学
二级分类:Algebraic Geometry 代数几何
分类描述:Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology
代数簇,叠,束,格式,模空间,复几何,量子上同调
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英文摘要:
We use Gale duality for polynomial complete intersections and adapt the proof of the fewnomial bound for positive solutions to obtain the bound (e^4+3) 2^(k choose 2) n^k/4 for the number of non-zero real solutions to a system of n polynomials in n variables having n+k+1 monomials whose exponent vectors generate a subgroup of Z^n of odd index. This bound exceeds the bound for positive solutions only by the constant factor (e^4+3)/(e^2+3) and it is asymptotically sharp for k fixed and n large.
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PDF链接:
https://arxiv.org/pdf/0706.4134