The Complex Conjugate Root Theorem states that if a polynomial has real coefficients and a complex number \( a + bi \) (where \( a \) and \( b \) are real numbers and \( i \) is the imaginary unit) as a root, then its complex conjugate \( a - bi \) must also be a root of the polynomial.
The Factor Theorem is a fundamental principle in algebra that relates to polynomials. It provides a way to determine whether a given polynomial has a particular linear factor. Specifically, the theorem states: If \( f(x) \) is a polynomial and \( c \) is a constant, then \( (x - c) \) is a factor of \( f(x) \) if and only if \( f(c) = 0 \).
The Grace–Walsh–Szegő theorem is a significant result in complex analysis and polynomial theory, particularly concerning the behavior of polynomials and their roots. The theorem deals with the location of the roots of a polynomial \( P(z) \) in relation to the roots of another polynomial \( Q(z) \). Specifically, it provides conditions under which all roots of \( P(z) \) lie within the convex hull of the roots of \( Q(z) \).
Hilbert's irreducibility theorem is a result in algebraic number theory, specifically related to the behavior of certain types of polynomial equations. Formulated by David Hilbert in the early 20th century, the theorem provides a significant insight into the irreducibility of polynomials over number fields.
Kharitonov's theorem is a result in control theory, particularly in the study of linear time-invariant (LTI) systems and the stability of polynomial systems. It is often used in the analysis of systems with polynomials that have parameters, allowing for the examination of how variations in those parameters affect stability. The theorem provides a method to determine the stability of a family of linear systems defined by a parameterized characteristic polynomial.
Marden's theorem is a result in complex analysis that deals with the roots of a polynomial and their geometric properties, particularly concerning the locations of the roots in the complex plane.
Mason–Stothers theorem is a result in complex analysis and the theory of meromorphic functions, specifically concerning the growth and distribution of the zeros of these functions. It is a generalization of the classical results about the growth of entire functions and provides a way to relate the growth of a meromorphic function to the distribution of its zeros and poles.
The Polynomial Remainder Theorem is a fundamental result in algebra that relates to the division of polynomials. It states that if a polynomial \( f(x) \) is divided by a linear polynomial of the form \( (x - c) \), the remainder of this division is equal to the value of the polynomial evaluated at \( c \).
The Rational Root Theorem is a useful tool in algebra for finding the possible rational roots of a polynomial equation. It states that if a polynomial \( P(x) \) with integer coefficients has a rational root \( \frac{p}{q} \) (in lowest terms), where \( p \) and \( q \) are integers, then: - \( p \) (the numerator) must be a divisor of the constant term of the polynomial.
The Friedlander–Iwaniec theorem is a result in number theory, specifically in the area of additive number theory concerning the distribution of prime numbers. It was established by the mathematicians J. Friedlander and H. Iwaniec in the early 1990s.
The Gap Theorem is a concept in the field of mathematics, particularly in the study of algebraic geometry and topology, though there are applications and related ideas in other areas of mathematics as well. In one of its forms, the Gap Theorem refers to a result concerning the existence of "gaps" in the spectrum of certain types of operators, particularly in the context of spectral theory.
The Geiringer–Laman theorem is a result in the field of graph theory and combinatorial geometry, specifically concerning the rigidity of frameworks. The theorem provides a criterion for determining when a certain kind of graph, known as a "framework", can be considered rigid, meaning that its vertices cannot be moved without distorting the distances between them.
Abhyankar's lemma is a result in the area of algebraic geometry, specifically dealing with the properties of algebraic varieties and their points over fields. Named after the mathematician Shivaramakrishna Abhyankar, the lemma provides a criterion for the existence of certain types of points in the context of algebraic varieties defined over a field.
The Kronecker limit formula is an important result in the theory of modular forms and number theory. It relates the behavior of certain L-functions to the special values of those functions at integers. Specifically, it provides a way to compute the special value of an L-function associated with a point on a certain modular curve. The formula can be stated in the context of the Dedekind eta function and the Eisenstein series.
The Landau prime ideal theorem is a result in the field of algebra, specifically in commutative algebra and the theory of rings. It concerns the structure of prime ideals in a non-zero commutative ring.
The Landsberg–Schaar relation is a concept in the field of thermodynamics, particularly in relation to the thermoelectric properties of materials. It establishes a relationship between the electrical conductivity, the Seebeck coefficient, and the thermal conductivity of a material. This relation is significant because it helps to optimize materials for thermoelectric applications, such as in power generation or cooling devices.
Maier's theorem is a result in number theory related to the distribution of prime numbers. Specifically, it deals with the existence of certain arithmetic progressions among prime numbers. The theorem is typically discussed in the context of additive number theory and is named after the mathematician Helmut Maier, who contributed to the understanding of the distribution of primes.
The Petersson trace formula is an important result in the theory of modular forms and number theory. It provides a relationship between the eigenvalues of Hecke operators on modular forms and the values of L-functions at certain critical points. The formula is named after the mathematician Heinrich Petersson, who was instrumental in its development. In its most common form, the Petersson trace formula connects the spectral theory of automorphic forms with the arithmetic of numbers through the Fourier coefficients of modular forms.
Vinogradov's mean-value theorem is a result in additive number theory that concerns the distribution of the values of additive functions. It has significant implications for the study of Diophantine equations and is particularly important in the field of analytic number theory. The theorem essentially states that for a certain class of additive functions (typically of the type that can be exhibited as sums of integers), the average number of representations of a number as a sum of other integers can be understood in a mean-value sense.