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The problem for graphs is NP-complete if the edge lengths are assumed integers. The problem for points on the plane is NP-complete with the discretized Euclidean metric and rectilinear metric. The problem is known to be NP-hard with the (non-discretized) Euclidean metric. [3]: ND22, ND23
The answer to the decision problem for the existential theory of the reals, given this sentence as input, is the Boolean value true. The inequality of arithmetic and geometric means states that, for every two non-negative numbers x {\displaystyle x} and y {\displaystyle y} , the following inequality holds: x + y 2 ≥ x y . {\displaystyle ...
Every polynomial in one variable x with real coefficients can be uniquely written as the product of a constant, polynomials of the form x + a with a real, and polynomials of the form x 2 + ax + b with a and b real and a 2 − 4b < 0 (which is the same thing as saying that the polynomial x 2 + ax + b has no real roots).
Finding the real roots of a polynomial with real coefficients is a problem that has received much attention since the beginning of 19th century, and is still an active domain of research. Most root-finding algorithms can find some real roots, but cannot certify having found all the roots.
This shows that the root α j will be less stable if there are many roots α k close to α j, in the sense that the distance |α j − α k | between them is smaller than |α j |. Example. For the root α 1 = 1, the derivative is equal to 1/19! which is very small; this root is stable even for large changes in t.
The word problem for an algebra is then to determine, given two expressions (words) involving the generators and operations, whether they represent the same element of the algebra modulo the identities. The word problems for groups and semigroups can be phrased as word problems for algebras. [1]
It follows from the present theorem and the fundamental theorem of algebra that if the degree of a real polynomial is odd, it must have at least one real root. [2] This can be proved as follows. Since non-real complex roots come in conjugate pairs, there are an even number of them;
A polynomial function P(s) of a complex variable s is said to be Hurwitz if the following conditions are satisfied: P(s) is real when s is real. The roots of P(s) have real parts which are zero or negative. Hurwitz polynomials are important in control systems theory, because they represent the characteristic equations of stable linear systems.