Rational lines on diagonal hypersurfaces and subconvexity via the circle method
HTML articles powered by AMS MathViewer
- by Trevor D. Wooley;
- Trans. Amer. Math. Soc. 377 (2024), 2125-2147
- DOI: https://doi.org/10.1090/tran/9077
- Published electronically: December 12, 2023
- HTML | PDF | Request permission
Abstract:
Fix $k,s,n\in \mathbb {N}$, and consider non-zero integers $c_1,\ldots ,c_s$, not all of the same sign. Provided that $s\geqslant k(k+1)$, we establish a Hasse principle for the existence of lines having integral coordinates lying on the affine diagonal hypersurface defined by the equation $c_1x_1^k+\ldots +c_sx_s^k=n$. This conclusion surmounts the conventional convexity barrier tantamount to the square-root cancellation limit for this problem.References
- G. I. Arkhipov, V. N. Chubarikov, and A. A. Karatsuba, Trigonometric sums in number theory and analysis, De Gruyter Expositions in Mathematics, vol. 39, Walter de Gruyter GmbH & Co. KG, Berlin, 2004. Translated from the 1987 Russian original. MR 2113479, DOI 10.1515/9783110197983
- B. J. Birch, Homogeneous forms of odd degree in a large number of variables, Mathematika 4 (1957), 102–105. MR 97359, DOI 10.1112/S0025579300001145
- Jean Bourgain, Ciprian Demeter, and Larry Guth, Proof of the main conjecture in Vinogradov’s mean value theorem for degrees higher than three, Ann. of Math. (2) 184 (2016), no. 2, 633–682. MR 3548534, DOI 10.4007/annals.2016.184.2.7
- Julia Brandes, Forms representing forms and linear spaces on hypersurfaces, Proc. Lond. Math. Soc. (3) 108 (2014), no. 4, 809–835. MR 3198749, DOI 10.1112/plms/pdt044
- Julia Brandes and Kevin Hughes, On the inhomogeneous Vinogradov system, Bull. Aust. Math. Soc. 106 (2022), no. 3, 396–403. MR 4510129, DOI 10.1017/S0004972722000284
- Richard Brauer, A note on systems of homogeneous algebraic equations, Bull. Amer. Math. Soc. 51 (1945), 749–755. MR 13127, DOI 10.1090/S0002-9904-1945-08440-7
- Jörg Brüdern and Trevor D. Wooley, Subconvexity for additive equations: pairs of undenary cubic forms, J. Reine Angew. Math. 696 (2014), 31–67. MR 3276162, DOI 10.1515/crelle-2012-0115
- Jörg Brüdern and Trevor D. Wooley, An instance where the major and minor arc integrals meet, Bull. Lond. Math. Soc. 51 (2019), no. 6, 1113–1128. MR 4041016, DOI 10.1112/blms.12291
- J. Brüdern and T. D. Wooley, On Waring’s problem for larger powers, J. Reine Angew. Math. (ahead of print), DOI 10.1515/crelle-2023-0072, arXiv:2211.10380.
- T. Estermann, A new application of the Hardy-Littlewood-Kloosterman method, Proc. London Math. Soc. (3) 12 (1962), 425–444. MR 137677, DOI 10.1112/plms/s3-12.1.425
- Scott T. Parsell, The density of rational lines on cubic hypersurfaces, Trans. Amer. Math. Soc. 352 (2000), no. 11, 5045–5062. MR 1778504, DOI 10.1090/S0002-9947-00-02635-0
- Scott T. Parsell, Asymptotic estimates for rational linear spaces on hypersurfaces, Trans. Amer. Math. Soc. 361 (2009), no. 6, 2929–2957. MR 2485413, DOI 10.1090/S0002-9947-09-04821-1
- R. C. Vaughan, A new iterative method in Waring’s problem, Acta Math. 162 (1989), no. 1-2, 1–71. MR 981199, DOI 10.1007/BF02392834
- R. C. Vaughan, On Waring’s problem for cubes. II, J. London Math. Soc. (2) 39 (1989), no. 2, 205–218. MR 991656, DOI 10.1112/jlms/s2-39.2.205
- R. C. Vaughan, The Hardy-Littlewood method, 2nd ed., Cambridge Tracts in Mathematics, vol. 125, Cambridge University Press, Cambridge, 1997. MR 1435742, DOI 10.1017/CBO9780511470929
- R. C. Vaughan and T. D. Wooley, Further improvements in Waring’s problem. II. Sixth powers, Duke Math. J. 76 (1994), no. 3, 683–710. MR 1309326, DOI 10.1215/S0012-7094-94-07626-6
- Robert C. Vaughan and Trevor D. Wooley, Further improvements in Waring’s problem, Acta Math. 174 (1995), no. 2, 147–240. MR 1351319, DOI 10.1007/BF02392467
- Trevor D. Wooley, Vinogradov’s mean value theorem via efficient congruencing, Ann. of Math. (2) 175 (2012), no. 3, 1575–1627. MR 2912712, DOI 10.4007/annals.2012.175.3.12
- Trevor D. Wooley, The asymptotic formula in Waring’s problem, Int. Math. Res. Not. IMRN 7 (2012), 1485–1504. MR 2913181, DOI 10.1017/S030500410200628X
- Trevor D. Wooley, Discrete Fourier restriction via efficient congruencing, Int. Math. Res. Not. IMRN 5 (2017), 1342–1389. MR 3658168, DOI 10.1093/imrn/rnw031
- Trevor D. Wooley, The cubic case of the main conjecture in Vinogradov’s mean value theorem, Adv. Math. 294 (2016), 532–561. MR 3479572, DOI 10.1016/j.aim.2016.02.033
- Trevor D. Wooley, On Waring’s problem for intermediate powers, Acta Arith. 176 (2016), no. 3, 241–247. MR 3580113, DOI 10.4064/aa8439-8-2016
- Trevor D. Wooley, Nested efficient congruencing and relatives of Vinogradov’s mean value theorem, Proc. Lond. Math. Soc. (3) 118 (2019), no. 4, 942–1016. MR 3938716, DOI 10.1112/plms.12204
- Trevor D. Wooley, Subconvexity in the inhomogeneous cubic Vinogradov system, J. Lond. Math. Soc. (2) 107 (2023), no. 2, 798–817. MR 4549146, DOI 10.1112/jlms.12698
- Trevor D. Wooley, Subconvexity in inhomogeneous Vinogradov systems, Q. J. Math. 74 (2023), no. 1, 389–418. MR 4571633, DOI 10.1093/qmath/haac027
- T. D. Wooley, Subconvexity and the Hilbert-Kamke problem, submitted, 13pp; arXiv:2201.02699.
Bibliographic Information
- Trevor D. Wooley
- Affiliation: Department of Mathematics, Purdue University, 150 N. University Street, West Lafayette, Indiana 47907-2067
- MR Author ID: 292520
- ORCID: 0000-0002-8781-4706
- Email: twooley@purdue.edu
- Received by editor(s): May 4, 2023
- Received by editor(s) in revised form: September 12, 2023
- Published electronically: December 12, 2023
- Additional Notes: The author was supported by NSF grants DMS-1854398 and DMS-2001549
- © Copyright 2023 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 377 (2024), 2125-2147
- MSC (2020): Primary 11D45, 11D72, 11P55
- DOI: https://doi.org/10.1090/tran/9077
- MathSciNet review: 4744752