Soit
Let
@article{JTNB_2014__26_3_579_0, author = {Chang, Mei-Chu and Kerr, Bryce and Shparlinski, Igor E. and Zannier, Umberto}, title = {Elements of large order on varieties over prime finite fields}, journal = {Journal de th\'eorie des nombres de Bordeaux}, pages = {579--593}, publisher = {Soci\'et\'e Arithm\'etique de Bordeaux}, volume = {26}, number = {3}, year = {2014}, doi = {10.5802/jtnb.880}, mrnumber = {3320493}, language = {en}, url = {http://www.numdam.org/articles/10.5802/jtnb.880/} }
TY - JOUR AU - Chang, Mei-Chu AU - Kerr, Bryce AU - Shparlinski, Igor E. AU - Zannier, Umberto TI - Elements of large order on varieties over prime finite fields JO - Journal de théorie des nombres de Bordeaux PY - 2014 SP - 579 EP - 593 VL - 26 IS - 3 PB - Société Arithmétique de Bordeaux UR - http://www.numdam.org/articles/10.5802/jtnb.880/ DO - 10.5802/jtnb.880 LA - en ID - JTNB_2014__26_3_579_0 ER -
%0 Journal Article %A Chang, Mei-Chu %A Kerr, Bryce %A Shparlinski, Igor E. %A Zannier, Umberto %T Elements of large order on varieties over prime finite fields %J Journal de théorie des nombres de Bordeaux %D 2014 %P 579-593 %V 26 %N 3 %I Société Arithmétique de Bordeaux %U http://www.numdam.org/articles/10.5802/jtnb.880/ %R 10.5802/jtnb.880 %G en %F JTNB_2014__26_3_579_0
Chang, Mei-Chu; Kerr, Bryce; Shparlinski, Igor E.; Zannier, Umberto. Elements of large order on varieties over prime finite fields. Journal de théorie des nombres de Bordeaux, Tome 26 (2014) no. 3, pp. 579-593. doi : 10.5802/jtnb.880. http://www.numdam.org/articles/10.5802/jtnb.880/
[1] O. Ahmadi, I. E. Shparlinski and J. F. Voloch, Multiplicative order of Gauss periods, Intern. J. Number Theory, 6, (2010), 877–882. | MR | Zbl
[2] I. Aliev and C. J. Smyth, Solving algebraic equations in roots of unity, Forum Math., 24, (2012), 641–665. | MR | Zbl
[3] F. Beukers and C. J. Smyth, Cyclotomic points on curves, Number theory for the millenium (Urbana, Illinois, 2000), I, A.K. Peters, (2002), 67–85. | MR | Zbl
[4] E. Bombieri and W. Gubler, Heights in Diophantine geometry, Cambridge Univ. Press, Cambridge, (2006). | MR | Zbl
[5] J. Bourgain, M. Z. Garaev, S. V. Konyagin and I. E. Shparlinski, On the hidden shifted power problem, SIAM J. Comp., 41, (2012), 1524–1557. | MR
[6] J. F. Burkhart, N. J. Calkin, S. Gao, J. C. Hyde-Volpe, K. James, H. Maharaj, S. Manber, J. Ruiz and E. Smith, Finite field elements of high order arising from modular curve, Designs, Codes and Cryptography, 51, (2009), 301–314. | MR | Zbl
[7] M.-C. Chang, Order of Gauss periods in large characteristic, Taiwanese J. Math., 17, (2013), 621–628. | MR
[8] M.-C. Chang, Elements of large order in prime finite fields, Bull. Aust. Math. Soc, 88, (2013), 169–176. | MR | Zbl
[9] Q. Cheng, S. Gao and D. Wan, Constructing high order elements through subspace polynomials, Proc. 23rd ACM-SIAM Symposium on Discrete Algorithms, SIAM Press, (2012), 1457–1463. | MR
[10] D. A. Cox, J. Little and D. O’Shea, Ideals, varieties, and algorithms, Springer-Verlag, (1992). | MR | Zbl
[11] C. D’Andrea, T. Krick and M. Sombra, Heights of varieties in multiprojective spaces and arithmetic Nullstellensätze, Annales Sci. de l’ENS, 46, (2013), 549–627. | MR
[12] P. Erdős and R. Murty, On the order of
[13] Z. Dvir, J. Kollár and S. Lovett, Variety evasive sets, Comp. Complex., (to appear). | MR
[14] K. Ford, The distribution of integers with a divisor in a given interval, Annals Math., 168, (2008), 367–433. | MR | Zbl
[15] J. von zur Gathen and I. E. Shparlinski, Gauss periods in finite fields, Proc. 5th Conference of Finite Fields and their Applications, Augsburg, 1999, Springer-Verlag, Berlin, (2001), 162–177. | MR | Zbl
[16] T. Krick, L. M. Pardo and M. Sombra, Sharp estimates for the arithmetic Nullstellensatz, Duke Math. J., 109, (2001), 521–598. | MR | Zbl
[17] M. Laurent, Équations diophantiennes exponentielles, Invent. Math., 78, (1984), 299–327. | MR | Zbl
[18] L. Leroux, Computing the torsion points of a variety defined by lacunary polynomials, Math. Comp., 81, (2012), 1587–1607. | MR | Zbl
[19] R. Popovych, Elements of high order in finite fields of the form
[20] R. Popovych, Elements of high order in finite fields of the form
[21] V. Shoup, Searching for primitive roots in finite fields, Math. Comp., 58, (1992), 369–380. | MR | Zbl
[22] I. E. Shparlinski, On primitive elements in finite fields and on elliptic curves, Matem. Sbornik, 181, (1990), 1196–1206 (in Russian). | MR | Zbl
[23] I. E. Shparlinski, Approximate constructions in finite fields, Proc. 3rd Conf. on Finite Fields and Appl., Glasgow, 1995, London Math. Soc., Lect. Note Series, 233, (1996), 313–332. | MR | Zbl
[24] I. Shparlinski, On the multiplicative orders of
[25] J. F. Voloch, On the order of points on curves over finite fields, Integers, 7, (2007), Article A49, 4 pp. | MR | Zbl
[26] J. F. Voloch, Elements of high order on finite fields from elliptic curves, Bull. Aust. Math. Soc., 81, (2010), 425–429. | MR | Zbl
[27] U. Zannier, Lecture notes on Diophantine analysis, Publ. Scuola Normale Superiore, Pisa, (2009). | MR | Zbl
[28] U. Zannier, Some problems of unlikely intersections in arithmetic and geometry, Priceton Univ. Press, Priceton, (2012). | MR | Zbl
Cité par Sources :