实代数曲线的伴随不等式
Adjunction inequality for real algebraic curves
摘要：
二元实多项式的零点集是$\mathbb R^2$中的曲线。对于其系数的一般选择，这是一条非奇异曲线，是适当嵌入$\mathbb R^2$中的圆和线的集合。对于给定次数的多项式，这些圆和直线的拓扑排列是什么？这个问题出现在19世纪的作品Harnack和希尔伯特，并列入希尔伯特到他的第16个问题。从那时起，取得了一些部分成果。然而，完整的答案是已知的，只有多项式的次数为5或更少。本文提出了一个新的部分结果的解决方案的第16希尔伯特问题。证明利用了Kronheimer和Mrowka在$\mathbb C P^2$中对Thom猜想的证明。
英文摘要：
The zero set of a real polynomial in two variable is a curve in $\mathbb R^2$. For a generic choice of its coefficients this is a non-singular curve, a collection of circles and lines properly embedded in $\mathbb R^2$. What topological arrangements of these circles and lines appear for the polynomials of a given degree? This question arised in the 19th century in the works of Harnack and Hilbert and was included by Hilbert into his 16th problem. Several partial results were obtained since then. However the complete answer is known only for polynomials of degree 5 or less. The paper presents a new partial result toward the solution of the 16th Hilbert problem. The proof makes use of the proof by Kronheimer and Mrowka of the Thom conjecture in $\mathbb C P^2$.
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