Abstract
We prove a theorem on algebraic osculation and apply our result to the computer algebra problem of polynomial factorization. We consider X a smooth completion of ℂ2 and D an effective divisor with support the boundary ∂X=X∖ℂ2. Our main result gives explicit conditions that are necessary and sufficient for a given Cartier divisor on the subscheme \((|D|,\mathcal{O}_{D})\) to extend to X. These osculation criteria are expressed with residues. When applied to the toric setting, our result gives rise to a new algorithm for factoring sparse bivariate polynomials which takes into account the geometry of the Newton polytope.
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Notes
The genericity must be defined relative to some invariant, such as the cardinality of interior lattice points, or the volume.
References
F. Abu Salem, S. Gao, A.G.B. Lauder, Factoring polynomials via polytopes, in Proc. of ISSAC (2004), pp. 4–11.
M. Avendano, T. Krick, M. Sombra, Factoring bivariate sparse (lacunary) polynomials, J. Complex. 23, 193–216 (2007).
W.P. Barth, K. Hulek, C.A.M. Peters, A. Van De Ven, Compact Complex Surfaces, 2nd edn. (Springer, Berlin, 2004).
K. Belabas, M. van Hoeij, J. Kluners, A. Steel, Factoring polynomials over global fields, J. Théor. Nr. Bordx. 21(1), 15–39 (2009).
G. Chèze, Absolute polynomial factorization in two variables and the knapsack problem, in Proc. of ISSAC (2004), pp. 87–94.
G. Chèze, A. Galligo, From an approximate to an exact factorization, J. Symb. Comput. 41(6), 682–696 (2006).
G. Chèze, G. Lecerf, Lifting and recombination techniques for absolute factorization, J. Complex. 23(3), 380–420 (2007).
V. Danilov, The geometry of toric varieties, Russ. Math. Surv. 33, 97–154 (1978).
M. Elkadi, A. Galligo, M. Weimann, Towards toric absolute factorization, J. Symb. Comput. 44(9), 1194–1211 (2009).
W. Fulton, Introduction to Toric Varieties. Annals of Math. Studies (Princeton University Press, Princeton, 1993).
A. Galligo, D. Rupprecht, Irreducible decomposition of curves, J. Symb. Comput. 33, 661–677 (2002).
S. Gao, A.G.B. Lauder, Decomposition of polytopes and polynomials, Discrete Comput. Geom. 6(1), 89–124 (2001).
M.L. Green, Secant functions, the Reiss relation and its converse, Trans. Am. Math. Soc. 280(2), 499–507 (1983).
P.A. Griffiths, Variations on a theorem of Abel, Invent. Math. 35, 321–390 (1976).
P.A. Griffiths, J. Harris, Principles of Algebraic Geometry. Pure and Applied Mathematics (Wiley-Interscience, New York, 1978).
P.A. Griffiths, J. Harris, Residues and zero-cycles on algebraic varieties, Ann. Math. 128, 461–505 (1978).
G. Henkin, M. Passare, Abelian differentials on singular varieties and variation on a theorem of Lie-Griffiths, Invent. Math. 135, 297–328 (1999).
A.G. Khovansky, Newton polyhedra and toric varieties, Funct. Anal. Appl. 11, 56–67 (1977).
G. Lecerf, Improved dense multivariate polynomial factorization algorithms, J. Symb. Comput. 42, 477–494 (2007).
A.M. Ostrowski, On multiplication and factorization of polynomials. Lexicographic orderings and extreme aggregates of terms, Aequ. Math. 13, 201–228 (1975).
A. Tsikh, Multidimensional residues and their applications, Trans. Am. Math. Soc. 103 (1992).
J. von zur Gathen, J. Gerhard, Modern Computer Algebra, 1st edn. (Cambridge University Press, Cambridge, 1999).
M. Weimann, Trace et calcul résiduel: nouvelle version du théorème d’Abel-inverse et formes abéliennes, Ann. Toulouse 16(2), 397–424 (2007).
M. Weimann, An interpolation theorem in toric varieties, Ann. Inst. Fourier 58(4), 1371–1381 (2008).
M. Weimann, A lifting and recombination algorithm for rational factorization of sparse polynomials, J. Complex. 26(6), 608–628 (2010).
J.A. Wood, Osculation by algebraic hypersurfaces, J. Differ. Geom. 18, 563–573 (1983).
Acknowledgements
We would like to thank Michel Brion, Stéphane Druel, José Ignacio Burgos, and Martin Sombra for their helpful comments. We also thank Mohamed Elkadi and André Galligo, who suggested that we pay attention to the interplay between toric geometry and sparse polynomial factorization.
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Communicated by Teresa Krick.
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Weimann, M. Algebraic Osculation and Application to Factorization of Sparse Polynomials. Found Comput Math 12, 173–201 (2012). https://doi.org/10.1007/s10208-012-9114-z
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DOI: https://doi.org/10.1007/s10208-012-9114-z
Keywords
- Polynomial factorization
- Newton polytope
- Toric varieties
- Curves
- Line bundles
- Osculation
- Residue
- Cohomology
- Duality