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The theorem is used to find all rational roots of a polynomial, if any. It gives a finite number of possible fractions which can be checked to see if they are roots. If a rational root x = r is found, a linear polynomial ( x – r ) can be factored out of the polynomial using polynomial long division , resulting in a polynomial of lower degree ...
If f is a function that is meromorphic on the whole Riemann sphere, then it has a finite number of zeros and poles, and the sum of the orders of its poles equals the sum of the orders of its zeros. Every rational function is meromorphic on the whole Riemann sphere, and, in this case, the sum of orders of the zeros or of the poles is the maximum ...
The theorem is usually used to simplify the problem of locating zeros, as follows. Given an analytic function, we write it as the sum of two parts, one of which is simpler and grows faster than (thus dominates) the other part. We can then locate the zeros by looking at only the dominating part.
The location of the Riemann zeta function's zeros is of great importance in number theory. The prime number theorem is equivalent to the fact that there are no zeros of the zeta function on the Re(s) = 1 line. [16] It is also known that zeros do not exist in certain regions slightly to the left of the Re(s) = 1 line, known as zero-free regions.
In mathematics, Hilbert's Nullstellensatz (German for "theorem of zeros", or more literally, "zero-locus-theorem") is a theorem that establishes a fundamental relationship between geometry and algebra. This relationship is the basis of algebraic geometry. It relates algebraic sets to ideals in polynomial rings over algebraically closed fields.
In particular, when the number of sign changes is zero or one, then there are exactly zero or one positive roots. A linear fractional transformation of the variable makes it possible to use the rule of signs to count roots in any interval. This is the basic idea of Budan's theorem and the Budan–Fourier theorem. Repeated division of an ...
Risch called it a decision procedure, because it is a method for deciding whether a function has an elementary function as an indefinite integral, and if it does, for determining that indefinite integral. However, the algorithm does not always succeed in identifying whether or not the antiderivative of a given function in fact can be expressed ...
The Dirichlet L-function L(s, χ) = 1 − 3 −s + 5 −s − 7 −s + ⋅⋅⋅ (sometimes given the special name Dirichlet beta function), with trivial zeros at the negative odd integers. Let χ be a primitive character modulo q, with q > 1. There are no zeros of L(s, χ) with Re(s) > 1. For Re(s) < 0, there are zeros at certain negative ...