On ternary quadratic forms over the rational numbers

Amir Jafari; Farhood Rostamkhani

Czechoslovak Mathematical Journal (2022)

  • Volume: 72, Issue: 4, page 1105-1119
  • ISSN: 0011-4642

Abstract

top
For a ternary quadratic form over the rational numbers, we characterize the set of rational numbers represented by that form over the rational numbers. Consequently, we reprove the classical fact that any positive definite integral ternary quadratic form must fail to represent infinitely many positive integers over the rational numbers. Our proof uses only the quadratic reciprocity law and the Hasse-Minkowski theorem, and is elementary.

How to cite

top

Jafari, Amir, and Rostamkhani, Farhood. "On ternary quadratic forms over the rational numbers." Czechoslovak Mathematical Journal 72.4 (2022): 1105-1119. <http://eudml.org/doc/298927>.

@article{Jafari2022,
abstract = {For a ternary quadratic form over the rational numbers, we characterize the set of rational numbers represented by that form over the rational numbers. Consequently, we reprove the classical fact that any positive definite integral ternary quadratic form must fail to represent infinitely many positive integers over the rational numbers. Our proof uses only the quadratic reciprocity law and the Hasse-Minkowski theorem, and is elementary.},
author = {Jafari, Amir, Rostamkhani, Farhood},
journal = {Czechoslovak Mathematical Journal},
keywords = {ternary quadratic forms; Gauss reciprocity law; Hasse-Minkowski theorem},
language = {eng},
number = {4},
pages = {1105-1119},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On ternary quadratic forms over the rational numbers},
url = {http://eudml.org/doc/298927},
volume = {72},
year = {2022},
}

TY - JOUR
AU - Jafari, Amir
AU - Rostamkhani, Farhood
TI - On ternary quadratic forms over the rational numbers
JO - Czechoslovak Mathematical Journal
PY - 2022
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 72
IS - 4
SP - 1105
EP - 1119
AB - For a ternary quadratic form over the rational numbers, we characterize the set of rational numbers represented by that form over the rational numbers. Consequently, we reprove the classical fact that any positive definite integral ternary quadratic form must fail to represent infinitely many positive integers over the rational numbers. Our proof uses only the quadratic reciprocity law and the Hasse-Minkowski theorem, and is elementary.
LA - eng
KW - ternary quadratic forms; Gauss reciprocity law; Hasse-Minkowski theorem
UR - http://eudml.org/doc/298927
ER -

References

top
  1. Albert, A. A., 10.2307/2371130, Am. J. Math. 55 (1933), 274-292. (1933) Zbl0006.29004MR1506964DOI10.2307/2371130
  2. Conway, J. H., 10.5948/UPO9781614440253, The Carus Mathematical Monographs 26. Mathematical Association of America, Washington (1997). (1997) Zbl0885.11002MR1478672DOI10.5948/UPO9781614440253
  3. Cox, D. A., 10.1002/9781118400722, Pure and Applied Mathematics. A Wiley Series of Texts, Monographs, and Tracts. John Wiley & Sons, Hoboken (2013). (2013) Zbl1275.11002MR3236783DOI10.1002/9781118400722
  4. Dickson, L. E., 10.1090/S0002-9904-1927-04312-9, Bull. Am. Math. Soc. 33 (1927), 63-70 9999JFM99999 53.0133.04. (1927) MR1561323DOI10.1090/S0002-9904-1927-04312-9
  5. Doyle, G., Williams, K. S., A positive-definite ternary quadratic form does not represent all positive integers, Integers 17 (2017), Article ID A.41, 19 pages. (2017) Zbl1412.11065MR3708292
  6. Duke, W., Schulze-Pillot, R., 10.1007/BF01234411, Invent. Math. 99 (1990), 49-57. (1990) Zbl0692.10020MR1029390DOI10.1007/BF01234411
  7. Flath, D. E., 10.1090/chel/384.H, AMS, Providence (2018). (2018) Zbl1400.11001MR3837147DOI10.1090/chel/384.H
  8. Gupta, H., 10.1007/BF03049015, Proc. Indian Acad. Sci., Sect. A 13 (1941), 519-520. (1941) Zbl0063.01797MR0004816DOI10.1007/BF03049015
  9. Jones, B. W., Pall, G., 10.1007/BF02547347, Acta Math. 70 (1939), 165-191. (1939) Zbl0020.10701MR1555447DOI10.1007/BF02547347
  10. Kaplansky, I., 10.4064/aa-70-3-209-214, Acta Arith. 70 (1995), 209-214. (1995) Zbl0817.11024MR1322563DOI10.4064/aa-70-3-209-214
  11. Kaplansky, I., Linear Algebra and Geometry: A Second Course, Dover Publications, Mineola (2003). (2003) Zbl1040.15001MR2001037
  12. Lebesgue, V. A., Tout nombre impair est la somme de quatre carrés dont deux sont égaux, J. Math. Pures Appl. (2) 2 (1857), 149-152 French. (1857) 
  13. Legendre, A. M., Essai sur la théorie des nombres, Duprat, Paris (1798), French. (1798) Zbl1395.11003MR2859036
  14. Mordell, L. J., 10.1515/crll.1931.164.40, J. Reine Angew. Math. 164 (1931), 40-49. (1931) Zbl0001.12001MR1581249DOI10.1515/crll.1931.164.40
  15. Mordell, L. J., 10.1090/S0002-9904-1932-05373-3, Bull. Am. Math. Soc. 38 (1932), 277-282. (1932) Zbl0004.20004MR1562374DOI10.1090/S0002-9904-1932-05373-3
  16. Ono, K., Soundararajan, K., 10.1007/s002220050191, Invent. Math. 130 (1997), 415-454. (1997) Zbl0930.11022MR1483991DOI10.1007/s002220050191
  17. Ramanujan, S., On the expression of a number in the form a x 2 + b y 2 + c z 2 + d u 2 , Proc. Camb. Philos. Soc. 19 (1917), 11-21 9999JFM99999 46.0240.01. (1917) 
  18. Serre, J.-P., 10.1007/978-1-4684-9884-4, Graduate Texts in Mathematics 7. Springer, New York (1973). (1973) Zbl0256.12001MR0344216DOI10.1007/978-1-4684-9884-4
  19. Sun, Z.-W., 10.1007/s11425-015-4994-4, Sci. China, Math. 58 (2015), 1367-1396. (2015) Zbl1348.11031MR3353977DOI10.1007/s11425-015-4994-4
  20. Wu, H.-L., Sun, Z.-W., Arithmetic progressions represented by diagonal ternary quadratic forms, Available at https://arxiv.org/abs/1811.05855v1 (2018), 16 pages. (2018) 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.