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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 .
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 .
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 ...
At the remaining critical point (0, 0) the second derivative test is insufficient, and one must use higher order tests or other tools to determine the behavior of the function at this point. (In fact, one can show that f takes both positive and negative values in small neighborhoods around (0, 0) and so this point is a saddle point of f.)
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 ...
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
Specifically, if the eigenvalues all have real parts that are negative, then the system is stable near the stationary point. If any eigenvalue has a real part that is positive, then the point is unstable. If the largest real part of the eigenvalues is zero, the Jacobian matrix does not allow for an evaluation of the stability. [12]