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In mathematics, the discriminant of a polynomial is a quantity that depends on the coefficients and allows deducing some properties of the roots without computing them. More precisely, it is a polynomial function of the coefficients of the original polynomial. The discriminant is widely used in polynomial factoring, number theory, and algebraic ...
For a function f of three or more variables, there is a generalization of the rule shown above. In this context, instead of examining the determinant of the Hessian matrix, one must look at the eigenvalues of the Hessian matrix at the critical point.
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When the curve C is algebraic, that is when it is defined by a bivariate polynomial f, then the discriminant is a useful tool to compute the critical points. Here we consider only the projection π y {\displaystyle \pi _{y}} ; Similar results apply to π x {\displaystyle \pi _{x}} by exchanging x and y .
The discriminant of K is 49 = 7 2. Accordingly, the volume of the fundamental domain is 7 and K is only ramified at 7. In mathematics, the discriminant of an algebraic number field is a numerical invariant that, loosely speaking, measures the size of the (ring of integers of the) algebraic number field.
If the discriminant is negative, then the parabola opens in the opposite direction, never crossing the -axis, and the equation has no real roots; in this case the two complex-valued roots will be complex conjugates whose real part is the value of the axis of symmetry.
The discriminant Δ of the cubic is the square of = () (), where a is the leading coefficient of the cubic, and r 1, r 2 and r 3 are the three roots of the cubic. As Δ {\displaystyle {\sqrt {\Delta }}} changes of sign if two roots are exchanged, Δ {\displaystyle {\sqrt {\Delta }}} is fixed by the Galois group only if the Galois group is A 3 .
The determinant of the Hessian at is called, in some contexts, a discriminant. If this determinant is zero then x {\displaystyle \mathbf {x} } is called a degenerate critical point of f , {\displaystyle f,} or a non-Morse critical point of f . {\displaystyle f.}