Discriminants, Resultants, and Multidimensional Determinants by Israel M. Gelfand, Mikhail Kapranov, Andrei Zelevinsky

By Israel M. Gelfand, Mikhail Kapranov, Andrei Zelevinsky

"This e-book revives and drastically expands the classical idea of resultants and discriminants. lots of the major new result of the ebook were released previous in additional than a dozen joint papers of the authors. The ebook properly enhances those unique papers with many examples illustrating either previous and new result of the theory."―Mathematical Reviews

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Extra info for Discriminants, Resultants, and Multidimensional Determinants (Modern Birkhäuser Classics)

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Q4}. Since Kx, Ky are conics, no three of the qi are collinear. All the conics through ql . . . q4 form a 1-dimensional linear system (pencil)/~ ~ P 1. The conics Kz, z ~ lp, contain q l . . . q4 and also form a pencil. Hence Kz, z ~ Ip are all the conics through ql . . . q4. Note that lp does not contain any of the qi. Indeed, otherwise we would have A (x, qi, qi) = A (y, qi, qi) = A (qi, qi, qi) = 0 for some i. This means that qi E X and the line < x, y > -- lp is tangent to X at qi, contrary to our assumption.

Boole (see [Ca l ]), who found that the degree of the discriminant is kd k-1 . Beyond this, little is known about the discriminant A x (in this example). Except in the case of a binary or quadratic form, it is never written in an expanded form, in part because it is so cumbersome. For example, the discriminant of the ternary cubic form Y~i+j+k=3 aijk Xi YJ Zk has degree 12 as a polynomial in (aijk) (and it is well-known in the theory of modular forms). The expanded form of this polynomial contains 2040 terms (we would like to thank S.

Among these coordinates we choose those in t~ and t2 as local coordinates on X1 x X2 near x0. The coordinates from the remaining six sets are analytic functions of these local coordinates vanishing at the origin. 9) regarded as a function of local coordinates from tl and t2. By some abuse of notation we write u as u -- tl + t2 -k- U l "~- U2 "~- tt + tu + ut + uu, 44 Chapter 1. Projective Dual Varieties where each of the summands stands for a linear combination of the variables of the corresponding group.

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