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Almost-toric hypersurfaces


An almost-toric hypersurface is parameterized by monomials multiplied by polynomials in one extra variable. We determine the Newton polytope of such a hypersurface, and apply this to give an algorithm for computing the implicit polynomial equation.

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  1. Botbol, N., Dickenstein, A.: Implicitization of rational hypersurfaces via linear syzygies: a practical overview (2015). doi:10.1016/j.jsc.2015.09.001

  2. Busé, L., Cox, D., D’Andrea, C.: Implicitization of surfaces in \({\mathbb{P}}^3\) in the presence of base points. J. Algebra Appl. 2(02), 189–214 (2003)

  3. Cox, D.: Equations of parametric curves and surfaces via syzygies In: Symbolic computation: solving equations in algebra, geometry, and engineering, pp. 1–20. South Hadley (2000)

  4. Cox, D., Little, J., Schenck, H.: Toric Varieties. American Mathematical Society, Providence (2011)

  5. Cueto, M.: Tropical implicitization. Ph.D. thesis, University of California, Berkeley (2010)

  6. Ilten, N., Süss, H.: Polarized complexity-1 \(T\)-varieties. Mich. Math. J. 60(3), 561–578 (2011)

  7. Maclagan, D., Sturmfels, B.: Introduction to Tropical Geometry, Graduate Texts in Mathematics, vol. 161. American Mathematical Society, Providence (2015)

  8. Philippon, P., Sombra, M.: A refinement of the Bernštein–Kušnirenko estimate. Adv. Math. 218(5), 1370–1418 (2008)

  9. Sturmfels, B., Tevelev, J.: Elimination theory for tropical varieties. Math. Res. Lett. 15(3), 543–562 (2008)

  10. Sturmfels, B.: Gröbner Bases and Convex Polytopes, vol. 8. American Mathematical Society, Providence (1996)

  11. Sturmfels, B., Tevelev, J., Josephine, Y.: The Newton polytope of the implicit equation. Mosc. Math. J 7(2), 327–346 (2007)

  12. Sturmfels, B., Yu, J.: Tropical Implicitization and Mixed Fiber Polytopes, Software for Algebraic Geometry, pp. 111–131. Springer, Berlin (2008)

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Author information

Correspondence to Bo Lin.

Additional information

The author thanks Bernd Sturmfels for his guidance, Ralph Morrison for his insightful suggestions and Nathan Ilten for discussions. This work was supported by the Thematic Program of National Institute for Mathematical Sciences, Daejeon, Korea, which hosted the author in the summer of 2014.

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Lin, B. Almost-toric hypersurfaces. Beitr Algebra Geom 57, 343–359 (2016). https://doi.org/10.1007/s13366-015-0268-0

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  • Toric variety
  • Tropical geometry
  • Almost-toric

Mathematics Subject Classification

  • Primary 14J70
  • Secondary 14T05