{"title":"二次多项式空间中PCF参数的几何","authors":"Laura DeMarco, Niki Myrto Mavraki","doi":"10.2140/ant.2025.19.2163","DOIUrl":null,"url":null,"abstract":"<p>We study algebraic relations among postcritically finite (PCF) parameters in the family <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mrow><mi>f</mi></mrow><mrow><mi>c</mi></mrow></msub><mo stretchy=\"false\">(</mo><mi>z</mi><mo stretchy=\"false\">)</mo>\n<mo>=</mo> <msup><mrow><mi>z</mi></mrow><mrow><mn>2</mn></mrow></msup>\n<mo>+</mo>\n<mi>c</mi></math>. It is known that an algebraic curve in <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mi>ℂ</mi></mrow><mrow><mn>2</mn></mrow></msup></math> contains infinitely many PCF pairs <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mo stretchy=\"false\">(</mo><msub><mrow><mi>c</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>c</mi></mrow><mrow><mn>2</mn></mrow></msub><mo stretchy=\"false\">)</mo></math> 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 <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mi>ℂ</mi></mrow><mrow><mi>n</mi></mrow></msup></math> for any <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>n</mi>\n<mo>≥</mo> <mn>2</mn></math>. Consequently, we obtain uniform bounds on the number of PCF pairs on nonspecial curves in <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mi>ℂ</mi></mrow><mrow><mn>2</mn></mrow></msup></math> and the number of PCF parameters in real algebraic curves in <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>ℂ</mi></math>, depending only on the degree of the curve. We also compute the optimal bound for the general curve of degree <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>d</mi></math>. For <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>d</mi>\n<mo>=</mo> <mn>1</mn></math>, we prove that there are only finitely many nonspecial lines in <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mi>ℂ</mi></mrow><mrow><mn>2</mn></mrow></msup></math> containing more than two PCF pairs, and similarly, that there are only finitely many (real) lines in <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>ℂ</mi>\n<mo>=</mo> <msup><mrow><mi>ℝ</mi></mrow><mrow><mn>2</mn></mrow></msup></math> containing more than two PCF parameters. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":"9 1","pages":""},"PeriodicalIF":1.0000,"publicationDate":"2025-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Geometry of PCF parameters in spaces of quadratic polynomials\",\"authors\":\"Laura DeMarco, Niki Myrto Mavraki\",\"doi\":\"10.2140/ant.2025.19.2163\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>We study algebraic relations among postcritically finite (PCF) parameters in the family <math display=\\\"inline\\\" xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"><msub><mrow><mi>f</mi></mrow><mrow><mi>c</mi></mrow></msub><mo stretchy=\\\"false\\\">(</mo><mi>z</mi><mo stretchy=\\\"false\\\">)</mo>\\n<mo>=</mo> <msup><mrow><mi>z</mi></mrow><mrow><mn>2</mn></mrow></msup>\\n<mo>+</mo>\\n<mi>c</mi></math>. It is known that an algebraic curve in <math display=\\\"inline\\\" xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"><msup><mrow><mi>ℂ</mi></mrow><mrow><mn>2</mn></mrow></msup></math> contains infinitely many PCF pairs <math display=\\\"inline\\\" xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"><mo stretchy=\\\"false\\\">(</mo><msub><mrow><mi>c</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>c</mi></mrow><mrow><mn>2</mn></mrow></msub><mo stretchy=\\\"false\\\">)</mo></math> 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 <math display=\\\"inline\\\" xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"><msup><mrow><mi>ℂ</mi></mrow><mrow><mi>n</mi></mrow></msup></math> for any <math display=\\\"inline\\\" xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"><mi>n</mi>\\n<mo>≥</mo> <mn>2</mn></math>. Consequently, we obtain uniform bounds on the number of PCF pairs on nonspecial curves in <math display=\\\"inline\\\" xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"><msup><mrow><mi>ℂ</mi></mrow><mrow><mn>2</mn></mrow></msup></math> and the number of PCF parameters in real algebraic curves in <math display=\\\"inline\\\" xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"><mi>ℂ</mi></math>, depending only on the degree of the curve. We also compute the optimal bound for the general curve of degree <math display=\\\"inline\\\" xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"><mi>d</mi></math>. For <math display=\\\"inline\\\" xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"><mi>d</mi>\\n<mo>=</mo> <mn>1</mn></math>, we prove that there are only finitely many nonspecial lines in <math display=\\\"inline\\\" xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"><msup><mrow><mi>ℂ</mi></mrow><mrow><mn>2</mn></mrow></msup></math> containing more than two PCF pairs, and similarly, that there are only finitely many (real) lines in <math display=\\\"inline\\\" xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"><mi>ℂ</mi>\\n<mo>=</mo> <msup><mrow><mi>ℝ</mi></mrow><mrow><mn>2</mn></mrow></msup></math> containing more than two PCF parameters. </p>\",\"PeriodicalId\":50828,\"journal\":{\"name\":\"Algebra & Number Theory\",\"volume\":\"9 1\",\"pages\":\"\"},\"PeriodicalIF\":1.0000,\"publicationDate\":\"2025-09-14\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Algebra & Number Theory\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.2140/ant.2025.19.2163\",\"RegionNum\":1,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Algebra & Number Theory","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.2140/ant.2025.19.2163","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
Geometry of PCF parameters in spaces of quadratic polynomials
We study algebraic relations among postcritically finite (PCF) parameters in the family . It is known that an algebraic curve in contains infinitely many PCF pairs 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 for any . Consequently, we obtain uniform bounds on the number of PCF pairs on nonspecial curves in 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 . For , we prove that there are only finitely many nonspecial lines in containing more than two PCF pairs, and similarly, that there are only finitely many (real) lines in containing more than two PCF parameters.
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