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The discriminant of a polynomial of degree n is homogeneous of degree 2n − 2 in the coefficients. This can be seen in two ways. In terms of the roots-and-leading-term formula, multiplying all the coefficients by λ does not change the roots, but multiplies the leading term by λ.
The discriminant of K can be referred to as the absolute discriminant of K to distinguish it from the relative discriminant of an extension K/L of number fields. The latter is an ideal in the ring of integers of L , and like the absolute discriminant it indicates which primes are ramified in K / L .
Let D be the discriminant of the field, n be the degree of K over , and = be the number of complex embeddings where is the number of real embeddings.Then every class in the ideal class group of K contains an integral ideal of norm not exceeding Minkowski's bound
Computational geometry, however, does frequently use calculations related to determinants. [ 51 ] While the determinant can be computed directly using the Leibniz rule this approach is extremely inefficient for large matrices, since that formula requires calculating n ! {\displaystyle n!} ( n {\displaystyle n} factorial ) products for an n × n ...
Figure 1. Plots of quadratic function y = ax 2 + bx + c, varying each coefficient separately while the other coefficients are fixed (at values a = 1, b = 0, c = 0). A quadratic equation whose coefficients are real numbers can have either zero, one, or two distinct real-valued solutions, also called roots.
The conic sections described by this equation can be classified in terms of the value , called the discriminant of the equation. [13] Thus, the discriminant is − 4Δ where Δ is the matrix determinant | / / |. If the conic is non-degenerate, then: [14] if B 2 − 4AC < 0, the equation represents an ellipse; if A = C and B = 0, the equation ...
2. In geometry and linear algebra, denotes the cross product. 3. In set theory and category theory, denotes the Cartesian product and the direct product. See also × in § Set theory. · 1. Denotes multiplication and is read as times; for example, 3 ⋅ 2. 2. In geometry and linear algebra, denotes the dot product. 3.
This "part" is the so-called complex Hessian, which is the matrix (¯),. Note that if f {\displaystyle f} is holomorphic, then its complex Hessian matrix is identically zero, so the complex Hessian is used to study smooth but not holomorphic functions, see for example Levi pseudoconvexity .