A refinement of the Bernstein-Kusnirenko estimate Export

@article{citeulike:2847162, abstract = {A theorem of Kusnirenko and Bernstein (also known as the BKK theorem) shows that the number of isolated solutions in a torus to a system of polynomial equations is bounded above by the mixed volume of the Newton polytopes of the given polynomials, and this upper bound is generically exact. We improve on this result by introducing refined combinatorial invariants of polynomials and a generalization of the mixed volume of convex bodies: the mixed integral of concave functions. The proof is based on new techniques and results from relative toric geometry.}, author = {Philippon, Patrice and Sombra, Mart\'{\i}n}, citeulike-article-id = {2847162}, citeulike-linkout-0 = {http://dx.doi.org/10.1016/j.aim.2008.03.019}, citeulike-linkout-1 = {http://www.sciencedirect.com/science/article/B6W9F-4SG555H-1/1/39d5f80f68d1e8308c1b33b0e2799180}, day = {1}, doi = {10.1016/j.aim.2008.03.019}, journal = {Advances in Mathematics}, keywords = {algebraic\_geometry, fewnomials, index, tropical\_geometry}, month = {August}, number = {5}, pages = {1370--1418}, posted-at = {2008-05-30 13:04:08}, priority = {2}, title = {A refinement of the Bernstein-Kusnirenko estimate}, url = {http://dx.doi.org/10.1016/j.aim.2008.03.019}, volume = {218}, year = {2008} }

TY - JOUR ID - citeulike:2847162 L3 - citeulike-article-id:2847162 N2 - A theorem of Kusnirenko and Bernstein (also known as the BKK theorem) shows that the number of isolated solutions in a torus to a system of polynomial equations is bounded above by the mixed volume of the Newton polytopes of the given polynomials, and this upper bound is generically exact. We improve on this result by introducing refined combinatorial invariants of polynomials and a generalization of the mixed volume of convex bodies: the mixed integral of concave functions. The proof is based on new techniques and results from relative toric geometry. IS - 5 JF - Advances in Mathematics EP - 1418 TI - A refinement of the Bernstein-Kusnirenko estimate VL - 218 SP - 1370 KW - algebraic_geometry KW - fewnomials KW - index KW - tropical_geometry AU - Philippon, Patrice AU - Sombra, Martín PY - 2008/08/01/ UR - http://dx.doi.org/10.1016/j.aim.2008.03.019 ER -