Gelfand, I.M., Kapranov, M.M., Zelevinsky, A.V., Discriminants, Resultants and Multidimensional Determinants. first edition. Birkhauser, Boston 1994.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....= x i (c x) for some constant c . That is = L f (c x) Hence ( f (L f (c x) p 1) f (c x) p 1) c f (x) 0. Similarly using the inverse Legendre transformation, we have ( c . Hence the result. One can also nd an indirect proof of this in [ GKZ]. Because of this result, we can now de ne the Legendre transformation for any nite set of homogenous polynomials via the conormal bundle of their common zero set. This approach works particularly well for projective spaces and it is closely related to the dual variety construction. 4.2 ....

I.M. Gelfand, M. Kapranov, A. Zelevinsky, Discriminants, resultants and multidimensional determinants, Birkhauser (1994).

.... question of extending the notion of determinant to higher dimensional arrays (e.g. cubic matrices A ijk ) and proposed several answers, under the name hyperdeterminants [6, 7] The most sophisticated notion of hyperdeterminant has been the object of recent investigations, summarized in the book [11]. However, the simplest possible generalization of the determinant, de ned for a kth order tensor on an n dimensional space by the k tuple alternating sum (which vanishes for odd k) Det k (A) 1 ; k 2Sn ( 1 ) k ) A 1 (i) k (i) 1) has been almost forgotten. ....

I M Gelfand, M M Kapranov and A V Zelevinsky, Discriminants, Resultants and Multidimensional Determinants, Birkhauser, Boston, MA, 1994, 523 pp.

....construction follows the same process as in the previous section, except that the notion of degree is changed. We consider n 1 Laurent polynomials f 0 ; f n 2 L = K [t 1 ; t n ] and we replace the constrains on the degree by constrains on the support of the polynomials [42, 85, 35]: Let x a polytope A i Z and assume that the support of f i is in A i : f i = 2A i c i; t The support of p = c x is the set of 2 Z such that c 6= 0 20 We denote by A the Minkowski sum of these polytopes (A = A 0 A n ) to which we associate the toric ....

....when the polynomials f 1 ; f n intersect properly, in the underlying projective toric variety. In this case, we will say that the system f 1 ; f n is generic for this construction. In this case, the dimension of A is the mixed volume of the polytopes of f 1 ; f n [42]. Here, again we have the property that for generic systems f 1 ; f n with support respectively in A 1 ; A n , the set of monomials x is a basis of A = R= f 1 ; f n ) 74, 37] 21 3.4 Resultant over a unirational variety A natural extension of the toric case consists ....

I. Gelfand, M. Kapranov, and A. Zelevinsky, Discriminants, Resultants and Multidimensional Determinants, Boston, Birkhauser, 1994.

....the set of integral points in the intersection of two rational halfspaces. Recall, that in the given situation the multiplicity e(R) is counted by the normalized volume ( twice the usual volume) of the set theoretic difference of the convex hull of and the convex hull of n f0g (see Chapter 5. 3 of [GKZ]) This set is triangulated by triangles whose only integral points are the vertices, which consist of two neighboring minimal generators of and the origin (see Figure 3) 20 (0,0) 2,1) 1,1) 1,2) e(R) 2(1 2 1 2) 2 Figure 3: Computing the multiplicity of k[x y; xy; y x] Each of ....

I. .M. Gelfand, M.M. Kapranov and A.V. Zelevinsky: Discriminants, Resultants and Multidimensional Determinants, Birkhauser, Boston, 1994.

....(n ) different sparse permutation matrices of size n in dimension d. The sparse permutations were used by E. Pascal in 1900 to define higher dimensional determinants [6] An alternative definition of higher dimensional determinants was quite recently given by Gelfand, Kapranov, and Zelevinsky [4]. 1.2 A geometric property of permutation matrices The generalization suggested in the present paper will contain the sparse permutations but also others. It is motivated by the following elegant connection between permutation matrices and geometry (possibly due to Schubert) cf. the papers by ....

I. M. Gelfand, M. M. Kapranov, and A. V. Zelevinsky, Discriminants, Resultants and Multidimensional Determinants, Birkhauser, Boston, 1994.

....;w d Gamma1 (j) Delta sgn(w) where sgn(w) i=1 sgn(w i ) So, now we have determinants, and hence minors, in higher dimensions. Observe that the above notion of a higher dimensional determinant is not the same as the one proposed in the recent book by Gelfand, Kapranov and Zelevinsky [9]. However, in their book they mention our definition as having been studied during the second half of the 19th century for which they refer to Ernst Pascal s Die Determinanten [12] from 1900, and further references therein. Let the rank of a d dimensional matrix be the size of its biggest ....

I. M. Gelfand, M. M. Kapranov, and A. V. Zelevinsky, Discriminants, Resultants and Multidimensional Determinants, Birkhauser, Boston, 1994.

....;w d Gamma1 (j) Delta sgn(w) where sgn(w) i=1 sgn(w i ) So, now we have determinants, and hence minors, in higher dimensions. Observe that the above notion of a higher dimensional determinant is not the same as the one proposed in the recent book by Gelfand, Kapranov and Zelevinsky [9]. However, in their book they mention our definition as having been studied during the second half of the 19th century for which they refer to Ernst Pascal s Die Determinanten [11] from 1900, and further references therein. Let the rank of a d dimensional matrix be the size of its biggest ....

I. M. Gelfand, M. M. Kapranov, and A. V. Zelevinsky, Discriminants, Resultants and Multidimensional Determinants, Birkhauser, Boston, 1994.

.... M p ffi F 00 : Remark 2 We can extend easily the construction of the map S to the case where the number of polynomials p 0 ; pm is greater than n 1 (m n) Operators of this type have been extensively used in the literature, in order, for instance, to dene resultants (see [24] 42] [18]) Let us recall that the vanishing of the resultant is the necessary and suOEcient condition on the coeOEcients of the polynomials p 0 ; pn , under which these polynomials have a common root (in a projective variety X) Two main examples appear in the literature: ffl The classical case ....

....resultant of the polynomials p 0 ; pn . The polynomials can also be homogenized in a toric sense, and the vanishing of the resultant is a necessary and suOEcient condition on their coeOEcients under which the toric homogenized polynomials have a common zero in the toric variety T (see [18]) We refer to this case as the toric case. Let us describe more carefully the monomials with exponents in F i used in the construction of the map S. The Macaulay case Let us x integers d 0 ; dn , and = d 0 Delta Delta Delta dn Gamma n. For any d 2 N, let R d denote the set of ....

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I. Gelfand, M. Kapranov, and A. Zelevinsky, Discriminants, Resultants and Multidimensional Determinants, Birkh#user, Boston-BaselBerlin, 1994.

....approach is related to the resultants of n 1 polynomials f 0 ; fn in n variables. The vanishing of the resultant over a projective variety X of these polynomials is the necessary and suOEcient condition on the coeOEcients of the polynomials f 0 ; fn to have a common root in X (see [9]) Our presentation 8 unies several known approaches under the same terminology of Sylvester map. In particular, we will cover the cases where X = P n is the projective space of dimension n, which yields the classical resultant (see [15] 28] and where X is a toric variety, which yields the ....

....known approaches under the same terminology of Sylvester map. In particular, we will cover the cases where X = P n is the projective space of dimension n, which yields the classical resultant (see [15] 28] and where X is a toric variety, which yields the so called toric resultant (see [9], 26] 2] The resultant can be computed as a divisor of the determinant of a map, which generalizes the Sylvester map for two polynomials in one variable. Let V 0 ; Vn be the n 1 vector spaces generated by monomials x E i = fx ff ; ff 2 E i g, where E i is the set of the ....

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I.M. Gelfand, M.M. Kapranov, and A.V. Zelevinsky. Discriminants, Resultants and Multidimensional Determinants. Birkh#user, Boston-BaselBerlin, 1994.

....context, the open subset U is (K f0g) m and the i;j are (Laurent) monomials in t 1 1 ; t 1 m . This yields to the notion of toric resultant, which is a condition on c such that the system f c homogenized in a convenient way has a solution in the corresponding toric variety [GKZ94], Stu93] CE93] Cox95] A resultant over a unirational algebraic variety is constructed in [BEM00] If X is a projective variety parameterized by a map de ned on an open subset U A m , and i;j are homogeneous polynomials such that i;j = i;j . The existence of an irreducible ....

....polynomial of degree d j k i . We denote by H the matrix (h i;j ) 1 i n;0 j m so that F = GH. We are looking for the condition(s) on the coecients c = c i;j ) of h i;j such that f c has a solution outside the variety V (G) de ned by G. In the next section, we extend the condition given in [GKZ94], for the existence of the resultant of global sections of m 1 invertible sheaves L 0 ; Lm on a projective variety X of dimension m. We show that the associated divisor is reduced and give its degree in terms of the rst Chern class of L 0 ; Lm . We use this generalization to ....

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I.M. Gelfand, M.M. Kapranov, and A.V. Zelevinsky. Discriminants, Resultants and Multidimensional Determinants. Boston, Birkhauser, 1994.

....of 1 2 d in V ol( 1 Q 1 d Q d ) Theorem 2.1 (BKK bound) Bernstein 75) Given a polynomial system P = ff 1 ; f d g and Q i = N (f i ) the number of solutions P has in (C ) d is equal to (Q 1 ; Q d ) for most choices of the coecients of P. See [GKZ94] for a full treatment of the subject. There are many ways to construct resultant matrices. The two most popular ones are (i) Sylvester s dialytic method, in which polynomials are multiplied by power products, and (ii) Cayley Dixon s discrete di erential method. De nition 2.6 Given a polynomial ....

I.M. Gelfand, M.M. Kapranov, and A.V. Zelevinsky. Discriminants, Resultants and Multidimensional Determinants. Birkhauser, Boston, rst edition, 1994. 13

.... systems and computing a projection operator from which the resultant can be extracted [12] This method has been experimentally found to be superior in performance on a wide variety of examples, in comparison with other elimination methods including Macaulay resultants, sparse resultants [3, 14, 8], the characteristic set construction [4] and the Gr obnerbasis construction [1, 2] The method takes less time, less space, as well as the extraneous factors seem to be fewer (except in the case of the Gr obnerbasis method which gives the exact resultant) 9] We also proved that for the unmixed ....

Gelfand, I., Kapranov, M., and Zelevinsky, A. Discriminants, Resultants and Multidimensional Determinants, rst ed. Birkhauser, Boston, 1994.

...., Y 1 ; Ym , Z 1 ; Z p . We will assume throughout that n m p. Let A = X 1in 1jm 1kp a ijk X i Y j Z k be a trilinear form in R, and let JA denote the ideal of R generated by all the partial derivatives of A. A question that arises from the theory of hyperdeterminants (see [GKZ, page 445]) is the following: What can be said about the ideal JA A reason for this question emerges, among other things, from results which show that information on the depth of JA and, more finely, on the primary decomposition of JA , is linked to information on the hyperdeterminant of A, see [BW] ....

I. M. Gelfand, M. M. Kapranov and A. V. Zelevinsky, Discriminants, Resultants and Multidimensional Determinants, Birkhauser, Boston, 1994.

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Gelfand, I.M., Kapranov, M.M., Zelevinsky, A.V., Discriminants, Resultants and Multidimensional Determinants. first edition. Birkhauser, Boston 1994.

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I. M. Gelfand, M. M. Kapranov, and A. V. Zelevinsky. Discriminants, Resultants and Multidimensional Determinants. Birkhauser, Boston, 1994.

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I.M. Gelfand, M.M. Kapranov, and A.V. Zelevinsky. Discriminants, Resultants and Multidimensional Determinants. Mathematics: Theory & Applications. Birkhauser, 1994. 127

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I.M. Gelfand, M.M. Kapranov, and A.V. Zelevinsky. Discriminants, Resultants and Multidimensional Determinants. Boston, Birkhauser, 1994.

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I.M. Gelfand, M.M. Kapranov, and A.V. Zelevinsky. Discriminants, Resultants and Multidimensional Determinants. Boston, Birkhauser, 1994.

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I.M. Gelfand, M.M. Kapranov, and A.V. Zelevinsky. Discriminants, Resultants and Multidimensional Determinants. Boston, Birkhauser, 1994.

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Gelfand, I., Kapranov, M., and Zelevinsky, A. Discriminants, Resultants and Multidimensional Determinants. Boston, Birkh#user, 1994.

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I.M. Gelfand, M.M. Kapranov, and A.V. Zelevinsky. Discriminants, Resultants and Multidimensional Determinants. Birkhauser, BostonBasel -Berlin, 1994.

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I.M. Gelfand, M.M. Kapranov, and A.V. Zelevinsky. Discriminants, Resultants and Multidimensional Determinants. Boston, Birkhauser, 1994.

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I.M. Gelfand, M.M. Kapranov, and A.V. Zelevinsky. Discriminants, Resultants and Multidimensional Determinants. Birkhauser, Boston, 1994.

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I. Gelfand, M. Kapranov, and A. Zelevinsky. Discriminants, Resultants and Multidimensional Determinants. Birkhauser Verlag, 1994.

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I.M. Gelfand, M. Kapranov, A. Zelevinsky, Discriminants, resultants and multidimensional determinants, Birkhauser (1994).

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