Number Theoretic Properties of Preperiodic Points
Let F(x) be a rational function of degree at least two with coefficients in Q. A theorem of Northcott says that F has only finitely many Q-rational preperiodic points, i.e., F has only finitely many preperiodic points in P1(Q). The Uniform Boundedness Conjecture of Morton and Silverman says that the number of preperiodic points of F in P1(Q) is bounded by a constant that depends only on the degree of F.
More generally, let F : PN → PN be a morphism of degree at least two defined over a number field K. Northcott's theorem says that F has only finitely many preperiodic points in PN(K), and the general Uniform Boundedness Conjecture says that the number of preperiodic points in PN(K) may be bounded solely in terms of N, the degree of F, and the degree of K over Q.
The Uniform Boundedness Conjecture is not known even for quadratic polynomials Fc(x) = x2+c over the rational numbers Q. It is known in this case that Fc(x) cannot have periodic points of period four, five, or six, although the result for period six is contingent on the validity of the conjecture of Birch and Swinnerton-Dyer. Poonen has conjectured that Fc(x) cannot have rational periodic points of any period strictly larger than three.
Read more about this topic: Arithmetic Dynamics
Famous quotes containing the words number, properties and/or points:
“Computers are good at swift, accurate computation and at storing great masses of information. The brain, on the other hand, is not as efficient a number cruncher and its memory is often highly fallible; a basic inexactness is built into its design. The brain’s strong point is its flexibility. It is unsurpassed at making shrewd guesses and at grasping the total meaning of information presented to it.”
—Jeremy Campbell (b. 1931)
“The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
—John Locke (1632–1704)
“The three main medieval points of view regarding universals are designated by historians as realism, conceptualism, and nominalism. Essentially these same three doctrines reappear in twentieth-century surveys of the philosophy of mathematics under the new names logicism, intuitionism, and formalism.”
—Willard Van Orman Quine (b. 1908)