Abstract
We present a uniform description of sets of m linear forms in n variables over the field of rational numbers whose computation requires m(n – 1) additions.
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A. V. Aho, J. E. Hopcroft and J. D. Ulhnan, The design and analysis of computer algorithms, Addison-Wesley, Reading, MA, 1974.
P. Bürgisser, M. Clausen and A. Shokrollahi, Algebraic complexity theory, Springer, Berlin, 1997.
L. Fukshansky, ‘Integral points of small height outside of a hypersurface’, Monatsh. Math., 147 (2006), 25–41.
J. Hadamard, ’Sur le module maximum que puisse atteindre un déterminant. C. R. Acad. Sci Paris 141 (1893), 1500–1501.
J. Heintz and C.-P. Schnorr, ’Testing polynomials which are easy to compute’. Int. Symp. on Logic and Algorithmic, Zürich 1980, Monogr. L’Enseign. Math., v. 30, 1982, Univ. Genéve, 237–254. Preliminary version in Proc. 12th ACM Symposium on Theory of Computing, 1980 , 263–272.
M. Kaminski and I. E. Shparlinski, ‘Sets of linear forms which are hard to compute’, Proc. 46th Intern. Symp. on Math. Found. of Comp. Sci. (MFCS), Schloss Dagstuhl - Leibniz-Zentrum LIPIcs, vol. 202, F. Bonchi and S.J. Puglisi, eds., Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021, 66:1–66:22.
P. Koiran, ‘Elimination of constants from machines over algebraically closed fields’, J. Compl., 13 (1997), 65–82.
T. Krick, L. M. Pardo and M. Sombra, ‘Sharp estimates for the arithmetic Nullstellensatz’, Duke Math. J., 109 3 (2001), 521–528.
L. Kronecker. ’Grundzüge einer arithmetischen Theorie der algebraischen Grössen’. J. reine angew. Math., 92 (1882), 1–122.
M. Mignotte, Mathematics for computer algebra, Springer-Verlag, Berlin, 1992.
O. Perron, Algebra I (Die Grundlagen), Walter de Gruyter, Berlin, 1927.
A. Ploski, ‘Algebraic dependence of polynomials after O. Perron and some applications’, Computational Commutative and Non-Commutative Algebraic Geometry, NATO Science Series, III: Computer and Systems Sciences, vol. 196, IOS Press, Amsterdam, 2005, 167–173.
J. E. Savage, ‘An algorithm for the computation of linear forms’, SIAM J. Comp., 3 (1974), 150–158.
J. T. Schwartz, ‘Fast probabilistic algorithms for verification of polynomial identities’, J. ACM, 27, (1980), 701–717.
A. Sert, ‘Une version effective du théorème de Lindemann–Weierstrass par les déterminants d’interpolation’, J. Number Theory, 76 (1999), 94–119.
V. Strassen, ‘Vermeidung von Divisionen’, J. reine angew. Math., 264 (1973), 184–202.
V. Strassen, ‘Polynomials with rational coefficients which are hard to compute’, SIAM J. Comp., 3 (1974), 128–149.
R. Zippel, ‘Probabilistic algorithms for sparse polynomials, Intern. Symp. on Symbolic and Algebraic Comp., 1979 , Lecture Notes in Computer Science, v.72, Springer, 1979, 216–226.
Acknowledgements
During the preparation of this work, I.E.S. was partially supported by ARC Grant DP200100355 as well as by a fellowship of the Foundation for Mathematical Sciences of Paris.
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Kaminski, M., Shparlinski, I.E. & Waldschmidt, M. On sets of linear forms of maximal complexity. comput. complex. 32, 1 (2023). https://doi.org/10.1007/s00037-022-00234-5
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DOI: https://doi.org/10.1007/s00037-022-00234-5