Geometry of PCF parameters in spaces of quadratic polynomials

IF 1 1区 数学 Q2 MATHEMATICS
Laura DeMarco, Niki Myrto Mavraki
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引用次数: 0

Abstract

We study algebraic relations among postcritically finite (PCF) parameters in the family fc(z) = z2 + c. It is known that an algebraic curve in 2 contains infinitely many PCF pairs (c1,c2) if and only if the curve is special (i.e., the curve is a vertical or horizontal line through a PCF parameter, or the curve is the diagonal). Here we extend this result to subvarieties of arbitrary dimension in n for any n 2. Consequently, we obtain uniform bounds on the number of PCF pairs on nonspecial curves in 2 and the number of PCF parameters in real algebraic curves in , depending only on the degree of the curve. We also compute the optimal bound for the general curve of degree d. For d = 1, we prove that there are only finitely many nonspecial lines in 2 containing more than two PCF pairs, and similarly, that there are only finitely many (real) lines in = 2 containing more than two PCF parameters.

二次多项式空间中PCF参数的几何
研究了fc(z)= z2+c族中后临界有限(PCF)参数之间的代数关系。已知一条代数曲线包含无穷多个PCF对(c1,c2),当且仅当该曲线是特殊的(即曲线是通过PCF参数的垂直线或水平线,或曲线是对角线)。这里我们将这个结果推广到对任意n≥2的任意维的子变种。因此,我们得到了在2中非特殊曲线上的PCF对的个数和在2中实代数曲线上的PCF参数的个数的一致界,它们只依赖于曲线的度。我们还计算了d次一般曲线的最优界。对于d= 1,我们证明了在2中包含两个以上PCF对的非特殊直线只有有限多条,同样,在= 2中包含两个以上PCF参数的(实)直线也只有有限多条。
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来源期刊
CiteScore
1.80
自引率
7.70%
发文量
52
审稿时长
6-12 weeks
期刊介绍: ANT’s inclusive definition of algebra and number theory allows it to print research covering a wide range of subtopics, including algebraic and arithmetic geometry. ANT publishes high-quality articles of interest to a broad readership, at a level surpassing all but the top four or five mathematics journals. It exists in both print and electronic forms. The policies of ANT are set by the editorial board — a group of working mathematicians — rather than by a profit-oriented company, so they will remain friendly to mathematicians'' interests. In particular, they will promote broad dissemination, easy electronic access, and permissive use of content to the greatest extent compatible with survival of the journal. All electronic content becomes free and open access 5 years after publication.
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