By Henri Darmon (auth.), Guillaume Hanrot, François Morain, Emmanuel Thomé (eds.)
This ebook constitutes the refereed lawsuits of the ninth overseas Algorithmic quantity idea Symposium, ANTS 2010, held in Nancy, France, in July 2010. The 25 revised complete papers offered including five invited papers have been rigorously reviewed and chosen for inclusion within the booklet. The papers are dedicated to algorithmic features of quantity thought, together with trouble-free quantity thought, algebraic quantity thought, analytic quantity idea, geometry of numbers, algebraic geometry, finite fields, and cryptography.
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Additional resources for Algorithmic Number Theory: 9th International Symposium, ANTS-IX, Nancy, France, July 19-23, 2010. Proceedings
Bernard and N. Gama f Ôx, y Õ aÔx ¡ yζf¡ ÕÔx ¡ yζf Õ where ζf¡ and ζf are the complex roots of the univariate polynomial f Ôx, 1Õ which we call the aﬃne representation of f . When Δf 0, each root of f live in RÞQ and the form is real. In this case, ζf¡ will denote the smallest root and ζf the largest one. When Δf 0, the roots are in CÞR and the form is imaginary. We note λÔf Õ min Ø f Ôx, y Õ : Ôx, y Õ È Z2 ÞÔ0, 0ÕÙ the ﬁrst minimum of f . We note Mt the transpose ¢ ofb aªmatrix M. The a ß2 polar representation of f is the symmetric matrix b of determinant ß2 c ¢ ª αβ ¡Δf ß4.
In: Boyd, C. ) ASIACRYPT 2001. LNCS, vol. 2248, pp. 480–494. Springer, Heidelberg (2001) 15. : Kedlaya’s algorithm in larger characteristic. Int Math Res Notices, Article ID No. rnm095, 2007, 29 (2007) 16. : Eﬃcient computation of p-adic heights. LMS J. Comput. Math. 11, 40–59 (2008) 17. : Counting points on hyperelliptic curves using Monsky-Washnitzer cohomology. J. Ramanujan Math. Soc. 16, 323–338 (2001); erratum ibid 18, 417– 418 (2003) 18. : The unipotent Albanese map and Selmer varieties for curves.
The reduction matrix from f to fr is ET ÔhÕ Ô0, 1; 1, hÕ. E. hÔg Õ, and ζg¡ h¡ ¡ 1 1 h ζ , c p Ô h Ôg ÕÕ Then we have h r g g g and pÔ0Õ a. By deﬁnition of h we have two cases: if ¡bß2c 0 then we have ¡bß2c, else we have ¡bß2c 0 h h h h¡ 0 g g . In both cases we Smallest Reduction Matrix of Binary Quadratic Forms pÔxÕ=cx2 +bx+a h pÔxÕ=cx2+bx+a a a h ζg¡ ζg 0 νg ¡b 2c Gauss ζg h=hÔg Õ νg cr ¡ 39 h-1 ζg h =hÔg Õ cr h-1 Red GL2 Fig. 1. Illustration of Lemma 2 This figure illustrates the convexity inequalities of Lemma 2.