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Combined effects of singular and exponential nonlinearities in fractional Kirchhoff problems
Orbit counting in polarized dynamical systems
Mathematics Department, Texas State University, San Marcos, TX 78666 |
We extend recent orbit counts for finitely generated semigroups acting on $ \mathbb{P}^N $ to certain infinitely generated, polarized semigroups acting on projective varieties. We then apply these results to semigroup orbits generated by some infinite sets of unicritical polynomials.
References:
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A. Baragar,
Rational points on $K3$ surfaces in $\mathbb{P}^1\times\mathbb{P}^1\times \mathbb{P}^1$, Math. Ann., 305 (1996), 541-558.
doi: 10.1007/BF01444236. |
| [2] |
E. Bombieri and J. Pila,
The number of integral points on arcs and ovals, Duke Math. J., 59 (1989), 337-357.
doi: 10.1215/S0012-7094-89-05915-2. |
| [3] |
Y. Bilu and R. Tichy,
The Diophantine equation $f(x) = g (y)$, Acta Arithmetica, 95 (2000), 261-288.
doi: 10.4064/aa-95-3-261-288. |
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W. Bosma, J. Cannon and C. Playoust,
The Magma algebra system I: The user language, J. Symbolic Comput., 24 (1997), 235-265.
doi: 10.1006/jsco.1996.0125. |
| [5] |
B. Brindza,
On $S$-integral solutions of the Catalan equation, Acta Arith., 48 (1987), 397-412.
doi: 10.4064/aa-48-4-397-412. |
| [6] |
G. Call and J. Silverman,
Canonical heights on varieties with morphisms, Compositio Math., 89 (1993), 163-205.
|
| [7] |
R. Crandall and C. Pomerance, Prime Numbers: A Computational Perspective, 2$^nd$ edition, Springer, New York, 2005. |
| [8] |
G. Faltings,
Endlichkeitssätze für abelsche Varietäten über Zahlkörpern, Invent. Math., 73 (1983), 349-366.
doi: 10.1007/BF01388432. |
| [9] |
P. Flajolet and R. Sedgewick, Analytic Combinatorics, Cambridge University Press, 2009.
|
| [10] |
D. R. Heath-Brown, Counting rational points on algebraic varieties, Analytic Number Theory, Lecture Notes in Math. Springer, Berlin, Heidelberg, (2006), 51–95.
doi: 10.1007/978-3-540-36364-4_2. |
| [11] |
D. R. Heath-Brown,
The density of rational points on curves and surfaces, Ann. of Math., 155 (2002), 553-598.
doi: 10.2307/3062125. |
| [12] |
V. Healey and W. Hindes,
Stochastic canonical heights, J. Number Theory, 201 (2019), 228-256.
doi: 10.1016/j.jnt.2019.02.020. |
| [13] |
W. Hindes, Counting points of bounded height in monoid orbits, preprint, arXiv: 2006.08563. |
| [14] |
W. Hindes, Dynamical and arithmetic degrees for random iterations of maps on projective space, Math. Proc. Camb. Philos. Soc, to appear. |
| [15] |
P. Ingram,
Lower bounds on the canonical height associated to the morphism, $\phi(z) = z^ d+ c$, Monatsh. Math., 157 (2009), 69-89.
doi: 10.1007/s00605-008-0018-6. |
| [16] |
S. Kawaguchi, Canonical heights for random iterations in certain varieties, Int. Math. Res. Not., Art. ID rnm 023, 2007 (2007), 33 pp. |
| [17] |
S. Kawaguchi and Joseph H. Silverman,
On the dynamical and arithmetic degrees of rational self-maps of algebraic varieties, J. Reine Angew. Math., 713 (2016), 21-48.
doi: 10.1515/crelle-2014-0020. |
| [18] |
R. Mason, Diophantine Equations Over Function Fields, London Mathematical Society Lecture Note Series, 96. Cambridge University Press, Cambridge, 1984.
doi: 10.1017/CBO9780511752490.
|
| [19] |
J. Mello,
On quantitative estimates for quasiintegral points in orbits of semigroups of rational maps, New York J. Math., 25 (2019), 1091-1111.
|
| [20] |
S. Schanuel,
On heights in number fields, Bull. Amer. Math. Soc., 70 (1964), 262-263.
doi: 10.1090/S0002-9904-1964-11110-1. |
| [21] |
D. Zagier,
On the number of Markoff numbers below a given bound, Math. Comp., 39 (1982), 709-723.
doi: 10.1090/S0025-5718-1982-0669663-7. |
| [22] |
Y. Zhang,
Bounded gaps between primes, Ann. of Math., 179 (2014), 1121-1174.
doi: 10.4007/annals.2014.179.3.7. |
show all references
References:
| [1] |
A. Baragar,
Rational points on $K3$ surfaces in $\mathbb{P}^1\times\mathbb{P}^1\times \mathbb{P}^1$, Math. Ann., 305 (1996), 541-558.
doi: 10.1007/BF01444236. |
| [2] |
E. Bombieri and J. Pila,
The number of integral points on arcs and ovals, Duke Math. J., 59 (1989), 337-357.
doi: 10.1215/S0012-7094-89-05915-2. |
| [3] |
Y. Bilu and R. Tichy,
The Diophantine equation $f(x) = g (y)$, Acta Arithmetica, 95 (2000), 261-288.
doi: 10.4064/aa-95-3-261-288. |
| [4] |
W. Bosma, J. Cannon and C. Playoust,
The Magma algebra system I: The user language, J. Symbolic Comput., 24 (1997), 235-265.
doi: 10.1006/jsco.1996.0125. |
| [5] |
B. Brindza,
On $S$-integral solutions of the Catalan equation, Acta Arith., 48 (1987), 397-412.
doi: 10.4064/aa-48-4-397-412. |
| [6] |
G. Call and J. Silverman,
Canonical heights on varieties with morphisms, Compositio Math., 89 (1993), 163-205.
|
| [7] |
R. Crandall and C. Pomerance, Prime Numbers: A Computational Perspective, 2$^nd$ edition, Springer, New York, 2005. |
| [8] |
G. Faltings,
Endlichkeitssätze für abelsche Varietäten über Zahlkörpern, Invent. Math., 73 (1983), 349-366.
doi: 10.1007/BF01388432. |
| [9] |
P. Flajolet and R. Sedgewick, Analytic Combinatorics, Cambridge University Press, 2009.
|
| [10] |
D. R. Heath-Brown, Counting rational points on algebraic varieties, Analytic Number Theory, Lecture Notes in Math. Springer, Berlin, Heidelberg, (2006), 51–95.
doi: 10.1007/978-3-540-36364-4_2. |
| [11] |
D. R. Heath-Brown,
The density of rational points on curves and surfaces, Ann. of Math., 155 (2002), 553-598.
doi: 10.2307/3062125. |
| [12] |
V. Healey and W. Hindes,
Stochastic canonical heights, J. Number Theory, 201 (2019), 228-256.
doi: 10.1016/j.jnt.2019.02.020. |
| [13] |
W. Hindes, Counting points of bounded height in monoid orbits, preprint, arXiv: 2006.08563. |
| [14] |
W. Hindes, Dynamical and arithmetic degrees for random iterations of maps on projective space, Math. Proc. Camb. Philos. Soc, to appear. |
| [15] |
P. Ingram,
Lower bounds on the canonical height associated to the morphism, $\phi(z) = z^ d+ c$, Monatsh. Math., 157 (2009), 69-89.
doi: 10.1007/s00605-008-0018-6. |
| [16] |
S. Kawaguchi, Canonical heights for random iterations in certain varieties, Int. Math. Res. Not., Art. ID rnm 023, 2007 (2007), 33 pp. |
| [17] |
S. Kawaguchi and Joseph H. Silverman,
On the dynamical and arithmetic degrees of rational self-maps of algebraic varieties, J. Reine Angew. Math., 713 (2016), 21-48.
doi: 10.1515/crelle-2014-0020. |
| [18] |
R. Mason, Diophantine Equations Over Function Fields, London Mathematical Society Lecture Note Series, 96. Cambridge University Press, Cambridge, 1984.
doi: 10.1017/CBO9780511752490.
|
| [19] |
J. Mello,
On quantitative estimates for quasiintegral points in orbits of semigroups of rational maps, New York J. Math., 25 (2019), 1091-1111.
|
| [20] |
S. Schanuel,
On heights in number fields, Bull. Amer. Math. Soc., 70 (1964), 262-263.
doi: 10.1090/S0002-9904-1964-11110-1. |
| [21] |
D. Zagier,
On the number of Markoff numbers below a given bound, Math. Comp., 39 (1982), 709-723.
doi: 10.1090/S0025-5718-1982-0669663-7. |
| [22] |
Y. Zhang,
Bounded gaps between primes, Ann. of Math., 179 (2014), 1121-1174.
doi: 10.4007/annals.2014.179.3.7. |
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