User:Richard Pinch/sandbox-WP

Cameron–Erdős conjecture

The Cameron-Erdős conjecture in the field of combinatorics within mathematics is the statement that the number of sum-free sets contained in \(\{1,\ldots,N\}\) is \(O\left({2^{N/2}}\right)\).

The conjecture was stated by Peter Cameron and Paul Erdős in 1988. It was proved by Ben Green in 2003 .

Conductor of an abelian variety

In mathematics, in Diophantine geometry, the conductor of an abelian variety defined over a local or global field F is a measure of how "bad" the bad reduction at some prime is. It is connected to the ramification in the field generated by the division points.

For an abelian variety A defined over a field F with ring of integers R, consider the Néron model of A, which is a 'best possible' model of A defined over R. This model may be represented as a scheme over

Spec(R)

(cf. spectrum of a ring) for which the generic fibre constructed by means of the morphism

Spec(F) → Spec(R)

gives back A. Let A⁰ denote the open subgroup scheme of the Néron model whose fibres are the connected components. For a residue field k, A⁰_k is a group variety over k, hence an extension of an abelian variety by a linear group. This linear group is an extension of a torus by a unipotent group. Let u be the dimension of the unipotent group and t the dimension of the torus. The order of the conductor is

\[ f = 2u + t + \delta , \, \]

where δ is a measure of wild ramification.

Properties

• If A has good reduction then f = u = t = δ = 0.
• If A has semistable reduction or, more generally, acquires semistable reduction over a Galois extension of F of degree prime to p, the residue characteristic, then δ = 0.
• If p > 2d + 1, where d is the dimension of A, then δ = 0.

References

• Template:Cite book
• Template:Cite journal

Cubic reciprocity

Various results connecting the solvability of two related cubic equations in modular arithmetic.

The law of cubic reciprocity is most naturally expressed in terms of the Eisenstein integers, that is, the ring $E$ of complex numbers of the form $$ \alpha = a + b\,\omega $$ where and $a$ and $b$ are integers and $\omega = \frac{1}{2}(-1 + i\sqrt 3) = e^{2\pi i/3}$ is a complex cube root of unity. The field norm $\mathbf{N}(\alpha) = \alpha \bar\alpha$.

If $\pi$ is a prime element of $E$ of norm $P$ and $\alpha$ is an element coprime to $\pi$, we define the cubic residue symbol $\left(\frac{\alpha}{\pi}\right)_3$ to be the cube root of unity (power of $\omega$) satisfying $$ \alpha^{(P-1)/3} \equiv \left(\frac{\alpha}{\pi}\right)_3 \pmod \pi $$

We further define a primary prime to be one which is congruent to $-1 \pmod 3$. Then for distinct primary primes $\pi$ and $\theta$ the law of cubic reciprocity is simply $$ \left(\frac{\pi}{\theta}\right)_3 = \left(\frac{\theta}{\pi}\right)_3 $$ with the supplementary laws for the units and for the prime $1-\omega$ of norm 3 that if $\pi = -1 + 3(m+n\omega)$ then $$ \left(\frac{\omega}{\pi}\right)_3 = \omega^{m+n}\,, $$ $$ \left(\frac{1-\omega}{\pi}\right)_3 = \omega^{2m} \ . $$

References

• David A. Cox, Primes of the form $x^2+ny^2$, Wiley, 1989, ISBN 0-471-50654-0.
• K. Ireland and M. Rosen, A classical introduction to modern number theory, 2nd ed, Graduate Texts in Mathematics 84, Springer-Verlag, 1990.
• Franz Lemmermeyer, Reciprocity laws: From Euler to Eisenstein, Springer Verlag, 2000, ISBN 3-540-66957-4.

Erdős–Fuchs theorem

In mathematics, in the area of combinatorial number theory, the Erdős–Fuchs theorem is a statement about the number of ways that numbers can be represented as a sum of two elements of a given set, stating that the average order of this number cannot be close to being a linear function.

The theorem is named after Paul Erdős and Wolfgang Heinrich Johannes Fuchs.

Statement

Let $A$ be a subset of the natural numbers and r(n) denote the number of ways that a natural number n can be expressed as the sum of two elements of A (taking order into account). We consider the average

\[R(n) = (r(1)+r(2)+\cdots+r(n) ) / n . \]

The theorem states that

\[R(n) = C + O\left(n^{-3/4-\epsilon}\right) \]

cannot hold unless C=0.

References

• Template:Cite journal
• Template:Cite book

Genus field

In algebraic number theory, the genus field $G$ of a number field $K$ is the maximal abelian extension of $K$ which is obtained by composing an absolutely abelian field with $K$ and which is unramified at all finite primes of $K$. The genus number of $K$ is the degree $[G:K]$ and the genus group is the Galois group of $G$ over $K$.

If $K$ is itself absolutely abelian, the genus field may be described as the maximal absolutely abelian extension of $K$ unramified at all finite primes: this definition was used by Leopoldt and Hasse.

If $K = \mathbb{Q}(\sqrt m)$ ($m$ squarefree) is a quadratic field of discriminant $D$, the genus field of $K$ is a composite of quadratic fields. Let $p_i$ run over the prime factors of $D$. For each such prime $p$, define $p^*$ as follows:

$$p^* = \pm p \equiv 1 \pmod 4 \text{ if } p \text{ is odd} ; $$ $$2^* = -4, 8, -8 \text{ according as } m \equiv 3 \pmod 4, 2 \pmod 8, -2 \pmod 8 . $$

Then the genus field is the composite $K(\sqrt{p^*_i})$.

See also

• Hilbert class field

References

• Template:Cite book

Group isomorphism problem

In abstract algebra, the group isomorphism problem is the decision problem of determining whether two group presentations present isomorphic groups.

The isomorphism problem was identified by Max Dehn in 1911 as one of three fundamental decision problems in group theory; the other two being the word problem and the conjugacy problem.

References

• Template:Cite book
• Template:Cite book
• Template:Cite journal

Hall algebra

The Hall polynomials in mathematics were developed by Philip Hall in the 1950s in the study of group representations.

A finite abelian p-group M is a direct sum of cyclic p-power components \(C_{p^\lambda_i}\) where \(\lambda=(\lambda_1,\lambda_2,\ldots)\) is a partition of \(n\) called the type of M. Let \(g^\lambda_{\mu,\nu}(p)\) be the number of subgroups N of M such that N has type \(\nu\) and the quotient M/N has type \(\mu\). Hall showed that the functions g are polynomial functions of p with integer coefficients: these are the Hall polynomials.

Hall next constructs an algebra \(H(p)\) with symbols \(u_\lambda\) a generators and multiplication given by the \(g^\lambda_{\mu,\nu}\) as structure constants

\[ u_\mu u_\nu = \sum_\lambda g^\lambda_{\mu,\nu} u_\lambda \]

which is freely generated by the \(u_{\mathbf1_n}\) corresponding to the elementary p-groups. The map from \(H(p)\) to the algebra of symmetric functions \(e_n\) given by \(u_{\mathbf 1_n} \mapsto p^{-n(n-1)}e_n\) is a homomorphism and its image may be interpreted as the Hall-Littlewood polynomial functions. The theory of Schur functions is thus closely connected with the theory of Hall polynomials.

References

• I.G. Macdonald, Symmetric functions and Hall polynomials, (Oxford University Press, 1979) ISBN 0-19-853530-9

Hall–Littlewood polynomials

In mathematics, the Hall–Littlewood polynomials encode combinatorial data relating to the representations of the general linear group. They are named for Philip Hall and Dudley E. Littlewood.

See also

• Littlewood polynomial

References

• I.G. Macdonald; Symmetric Functions and Hall Polynomials, (1979), pp. 101-104, Oxford University Press ISBN: 0-19-853530-9
• D.E. Littlewood; On certain symmetric functions, Proc. London Math. Soc., 43 (1961), pp. 485–498, DOI: 10.1112/plms/s3-11.1.485

Hutchinson operator

In mathematics, in the study of fractals, a Hutchinson operator is a collection of functions on an underlying space E. The iteration on these functions gives rise to an iterated function system, for which the fixed set is self-similar.

Definition

Formally, let f_i be a finite set of N functions from a set X to itself. We may regard this as defining an operator H on the power set P X as

\[H : A \mapsto \bigcup_{i=1}^N f_i[A],\,\]

where A is any subset of X.

A key question in the theory is to describe the fixed sets of the operator H. One way of constructing such a fixed set is to start with an initial point or set S₀ and iterate the actions of the f_i, taking S_n+1 to be the union of the images of S_n under the operator H; then taking S to be the union of the S_n, that is,

\[S_{n+1} = \bigcup_{i=1}^N f_i[S_n] \]

and

\[S = \bigcup_{n=0}^\infty S_n . \]

Properties

Hutchinson (1981) considered the case when the f_i are contraction mappings on a Euclidean space X = R^d. He showed that such a system of functions has a unique compact (closed and bounded) fixed set S.

The collection of functions \(f_i\) together with composition form a monoid. With N functions, then one may visualize the monoid as a full N-ary tree or a Cayley tree.

References

• Template:Cite journal
• Template:Cite book

Manin obstruction

In mathematics, in the field of arithmetic algebraic geometry, the Manin obstruction is attached to a geometric object X which measures the failure of the Hasse principle for X: that is, if the value of the obstruction is non-trivial, then X may have points over all local fields but not over a global field.

For abelian varieties the Manin obstruction is just the Tate-Shafarevich group and fully accounts for the failure of the local-to-global principle. There are however examples, due to Skorobogatov, of varieties with trivial Manin obstruction which have points everywhere locally and yet no global points.

References

• Serge Lang. "Survey of Diophantine geometry". (Springer-Verlag, 1997) ISBN 3-540-61223-8. Zbl 0869.11051. pp.250–258.
• Alexei Skorobogatov (1999). "Beyond the Manin obstruction" (with Appendix A by S. Siksek: 4-descent). Inventiones Mathematicae 135 no.2 (1999) 399–424. DOI 10.1007/s002220050291. Zbl 0951.14013.
• Alexei Skorobogatov (2001). "Torsors and rational points". Cambridge Tracts in Mathematics 144 (Cambridge: Cambridge University Press, 2001). ISBN 0-521-80237-7. Zbl 0972.14015. pp.1–7,112.

Pinch point

A pinch point or cuspidal point is a type of singular point on an algebraic surface. It is one of the three types of ordinary singularity of a surface.

The equation for the surface near a pinch point may be put in the form

\[ f(u,v,w) = u^2 - vw^2 + [4] \, \]

where [4] denotes terms of degree 4 or more.

References

• Template:Cite book

Z-number

Mahler's 3/2 problem concerns the existence of "Z-numbers".

A Z-number is a real number x such that the fractional parts $$ \left\lbrace x (3/2)^ n \right\rbrace $$ are less than 1/2 for all natural numbers $n$. Kurt Mahler conjectured in 1968 that there are no Z-numbers.

More generally, for a real number $\alpha$, define $\Omega(\alpha)$ as $$ \Omega(\alpha) = \inf_\theta\left({ \limsup_{n \rightarrow \infty} \left\lbrace{\theta\alpha^n}\right\rbrace - \liminf_{n \rightarrow \infty} \left\lbrace{\theta\alpha^n}\right\rbrace }\right) \ . $$ Mahler's conjecture would thus imply that $\Omega(3/2)$ exceeds 1/2. This is true, and indeed Flatto, Lagarias and Pollington showed that $$ \Omega(p/q) > 1/q $$ for rational $p/q$.

References

• Template:Cite book