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=Analytic subgroup theorem=
A significant result in modern [[transcendence theory]], a generalisation of [[Baker's theorem]] on linear forms in logarithms.

Let $G$ be a [[Abelian group|commutative]] [[algebraic group]] defined over a [[number field]] $K$ and let $B$ be a [[subgroup]] of the complex points $G(\mathbb{C})$ defined over $K$.  There are points of $B$ defined over the field of algebraic numbers if and only if there is a non-trivial [[analytic subgroup]] $H$ of $G$' defined over a number field such that $H(\mathbb{C})$ is contained in $B$.

==References==
* Alan Baker and Gisbert Wüstholz, ''Logarithmic Forms and Diophantine Geometry'', New Mathematical Monographs '''9''', Cambridge University Press (2007) ISBN 978-0-521-88268-2.  Chapter 6, pp.109-146.

=Conductor of an abelian variety=
In [[mathematics]], in [[Diophantine geometry]], the '''conductor of an abelian variety''' defined over a [[local field|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 point]]s.

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 (mathematics)|scheme]] over 

:Spec(''R'')

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

:Spec(''F'') &rarr; Spec(''R'')

gives back ''A''.  Let ''A''<sup>0</sup> denote the open subgroup scheme of the Néron model whose fibres are the connected components.  For a [[residue field]] ''k'', ''A''<sup>0</sup><sub>''k''</sub> 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

:<math> f = 2u + t + \delta , \, </math>

where &delta; is a measure of wild ramification.

==Properties==
* If ''A'' has [[good reduction]] then ''f'' = ''u'' = ''t'' = &delta; = 0.
* If ''A'' has [[semistable abelian variety|semistable reduction]] or, more generally, acquires semistable reduction over a Galois extension of ''F'' of degree prime to ''p'', the residue characteristic, then &delta; = 0.
* If ''p'' &gt; 2''d'' + 1, where ''d'' is the dimension of ''A'', then &delta; = 0.

==References==
* {{cite book | author=S. Lang | authorlink=Serge Lang | title=Survey of Diophantine geometry | publisher=[[Springer-Verlag]] | year=1997 | isbn=3-540-61223-8 | pages=70&ndash;71 }}
* {{cite journal | author=J.-P. Serre | coauthors=J. Tate | title=Good reduction of Abelian varieties | journal=Ann. Math. | volume=88 | year=1968 | pages=492&ndash;517 | doi=10.2307/1970722 | issue=3 | publisher=The Annals of Mathematics, Vol. 88, No. 3 | jstor=1970722 }}

=Critical exponent of a word=
In mathematics and computer science, the '''critical exponent''' of a finite or infinite sequence of symbols over a finite alphabet describes the largest number of times a contiguous subsequence can be repeated.  For example, the critical exponent of "Mississippi" is 7/3, as it contains the string "ississi", which is of length 7 and period 3.

If ''w'' is an infinite word over the alphabet ''A'' and ''x'' is a finite word over ''A'', then ''x'' is said to occur in ''w'' with exponent α, for positive real α, if there is a factor ''y'' of ''w'' with ''y'' = ''x''<sup>''a''</sup>''x''<sub>0</sub> where ''x''<sub>0</sub> is a prefix of ''x'', ''a'' is the integer part of α, and the length |''y''| ≥ α |''x''|: we say that ''y'' is an  ''α-power''.  The word ''w'' is  ''α-power-free'' if it contains no factors which are  α-powers.

The ''critical exponent'' for ''w'' is the [[supremum]] of the  α for which ''w'' has  α-powers, or equivalently the [[infimum]] of the  α for which ''w'' is  α-power-free.

==Examples==
* The critical exponent of the [[Fibonacci word]] is (5&nbsp;+&nbsp;√5)/2 ≈ 3.618.
* The critical exponent of the [[Thue–Morse sequence]] is 2. The word contains arbitrarily long squares, but in any factor ''xxb'' the letter ''b'' is not a prefix of ''x''.

==Properties==
* The critical exponent can take any real value greater than 1.
* The critical exponent of a [[morphic word]] over a finite alphabet is an [[algebraic number]] of degree at most the size of the alphabet.

==Repetition threshold==
The '''repetition threshold''' of an alphabet ''A'' of ''n'' letters is the minimum critical exponent of infinite words over ''A'': clearly this value RT(''n'') depends only on ''n''.  For ''n''=2, any binary word of length four has a factor of exponent 2, and since the critical exponent of the Thue–Morse sequence is 2, the repetition threshold for binary alphabets is RT(2) = 2.  It is known that RT(3) = 7/4, RT(4) = 7/5 and that for ''n''≥5 we have RT(''n'') ≥ ''n''/(''n''-1).  It is conjectured that the latter is the true value, and this has been established for 5 ≤ ''n'' ≤ 14 and for ''n'' ≥ 33.

==See also==
* [[Critical exponent]] of a physical system

==References==
* {{User:Richard_Pinch/sandbox/Ref | last1 = Allouche | first1 = Jean-Paul | last2 = Shallit | first2 = Jeffrey | author2-link = Jeffrey Shallit | isbn = 978-0-521-82332-6 | publisher = Cambridge University Press | title = Automatic Sequences: Theory, Applications, Generalizations | year = 2003 | zbl=1086.11015 }}
* {{User:Richard_Pinch/sandbox/Ref | last1=Berstel | first1=Jean | last2=Lauve | first2=Aaron | last3=Reutenauer | first3=Christophe | last4=Saliola | first4=Franco V. | title=Combinatorics on words. Christoffel words and repetitions in words | series=CRM Monograph Series | volume=27 | location=Providence, RI | publisher=American Mathematical Society | year=2009 | isbn=978-0-8218-4480-9 | url=http://www.ams.org/bookpages/crmm-27 | zbl=1161.68043 }}
* {{User:Richard_Pinch/sandbox/Ref  | title=Developments in Language Theory: Proceedings 10th International Conference, DLT 2006, Santa Barbara, CA, USA, June 26–29, 2006 | volume=4036 | series=Lecture Notes in Computer Science | editor1-first=Oscar H. | editor1-last=Ibarra | editor2-first=Zhe | editor2-last=Dang | publisher=Springer-Verlag | year=2006 | isbn=3-540-35428-X | first=Dalia | last=Krieger | chapter=On critical exponents in fixed points of non-erasing morphisms | pages=280–291 | zbl=1227.68074 }}
*{{User:Richard_Pinch/sandbox/Ref | first1=D. | last1=Krieger | first2=J. | last2=Shallit | title=Every real number greater than one is a critical exponent | journal=Theor. Comput. Sci. | volume=381 | year=2007 | pages=177–182 | doi=10.1016/j.tcs.2007.04.037 }}
* {{User:Richard_Pinch/sandbox/Ref  | last=Lothaire | first=M. | authorlink=M. Lothaire | title=Algebraic combinatorics on words | others=With preface by Jean Berstel and Dominique Perrin | edition=Reprint of the 2002 hardback | series=Encyclopedia of Mathematics and Its Applications | volume=90| publisher=Cambridge University Press | year=2011 | isbn=978-0-521-18071-9 | zbl=1221.68183 }}
* {{User:Richard_Pinch/sandbox/Ref  | last=Pytheas Fogg | first=N. | others=Editors Berthé, Valérie; Ferenczi, Sébastien; Mauduit, Christian; Siegel, A. | title=Substitutions in dynamics, arithmetics and combinatorics | series=Lecture Notes in Mathematics | volume=1794 | location=Berlin | publisher=Springer-Verlag | year=2002 | isbn=3-540-44141-7 | zbl=1014.11015 }}

=Cubic reciprocity=
Various results connecting the solvability of two related [[cubic equation]]s in [[modular arithmetic]]. 

The law of cubic reciprocity is most naturally expressed in terms of the [[Eisenstein integer]]s, that is, the ring $E$ of [[complex number]]s of the form 
$$
\alpha = a + b\,\omega
$$
where and $a$ and $b$ are [[integer]]s 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 [[Eisenstein prime|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 

:<math>R(n) = (r(1)+r(2)+\cdots+r(n) ) / n . </math>

The theorem states that 

:<math>R(n) = C + O\left(n^{-3/4-\epsilon}\right) </math>

cannot hold unless ''C''=0.


==References==
* {{cite journal | title=On a Problem of Additive Number Theory | author=P. Erdős | authorlink=Paul Erdős | coauthors=W.H.J. Fuchs | journal=Journal of the London Mathematical Society | year=1956 | volume=31 | issue=1 | pages=67–73 | doi=10.1112/jlms/s1-31.1.67 }}
* {{cite book | author=Donald J. Newman | title=Analytic number theory | series=[[Graduate Texts in Mathematics|GTM]] | volume=177 | year=1998 | isbn=0-387-98308-2 | pages=31-38 }}

=Genus field=
In [[algebraic number theory]], the '''genus field''' $G$ of a [[number field]] $K$ is the [[maximal]] [[abelian extension|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==
* {{cite book | zbl=0353.12001 | last=Ishida | first=Makoto | title=The genus fields of algebraic number fields | series=Lecture Notes in Mathematics | volume=555 | publisher=[[Springer-Verlag]] | year=1976 | isbn=3-540-08000-7 }}

=Group isomorphism problem=
In [[abstract algebra]], the '''group isomorphism problem''' is the [[decision problem]] of determining whether two [[Presentation of a group|group presentation]]s present [[Isomorphism|isomorphic]] [[Group (mathematics)|group]]s.

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 for groups|word problem]] and the [[conjugacy problem]].  

==References==
* {{cite book
  | last = Magnus
  | first = Wilhelm
  | authorlink = Wilhelm Magnus
  | coauthors = Abraham Karrass, Donald Solitar
  | title = Combinatorial group theory. Presentations of groups in terms of generators and relations
  | publisher = [[Dover Publications]]
  | date = 1976
  | location = 
  | pages = 24
  | url = 
  | doi = 
  | id = 
  | isbn = 0-486-63281-4}}
* {{cite book
  | last = Johnson
  | first = D.L.
  | authorlink = 
  | coauthors = 
  | title = Presentations of groups
  | publisher = [[Cambridge University Press]]
  | date = 1990
  | location = 
  | pages = 49
  | url = 
  | doi = 
  | id = 
  | isbn = 0-521-37203-8}}
* {{cite journal
  | last = Dehn
  | first = Max
  | authorlink = Max Dehn
  | coauthors = 
  | title = Über unendliche diskontinuierliche Gruppen
  | journal = [[Mathematische Annalen|Math. Ann.]]
  | volume = 71
  | issue = 
  | pages = 116-144
  | publisher = 
  | location = 
  | date = 1911
  | url = 
  | doi = 10.1007/BF01456932
  | id = 
  | accessdate = }}

=Hall algebra=

The '''Hall polynomials''' in [[mathematics]] were developed by [[Philip Hall]] in the 1950s in the study of [[group representation]]s.

A [[finite set|finite]] [[abelian group|abelian]] [[p-group|''p''-group]] ''M'' is a direct sum of [[cyclic group|cyclic]] ''p''-power components 
<math>C_{p^\lambda_i}</math> where
<math>\lambda=(\lambda_1,\lambda_2,\ldots)</math> is a [[Partition_%28number_theory%29|partition]] of <math>n</math> called the ''type'' of ''M''.  Let <math>g^\lambda_{\mu,\nu}(p)</math> be the number of
subgroups ''N'' of ''M'' such that ''N'' has type <math>\nu</math> and the quotient ''M/N'' has type <math>\mu</math>.  Hall showed that the functions ''g'' are [[polynomial]] functions of ''p'' with integer coefficients: these are the ''Hall polynomials''.

Hall next constructs an [[Algebra over a field|algebra]] <math>H(p)</math> with symbols <math>u_\lambda</math> a generators and multiplication given by the <math>g^\lambda_{\mu,\nu}</math> as structure constants

:<math> u_\mu u_\nu = \sum_\lambda g^\lambda_{\mu,\nu} u_\lambda </math>

which is freely generated by the <math>u_{\mathbf1_n}</math> corresponding to the elementary ''p''-groups.
The map from <math>H(p)</math> to the algebra of [[symmetric function]]s <math>e_n</math> given
by <math>u_{\mathbf 1_n} \mapsto p^{-n(n-1)}e_n</math> is a homomorphism and its image may be
interpreted as the [[Hall-Littlewood polynomial]] functions.  The theory of [[Schur function]]s 
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 [[Group representation|representation]]s of the [[general linear group]].  They are named for [[Philip Hall]] and [[Dudley E. Littlewood]].

==See also==
* [[Littlewood polynomial]]

==References==
* {{User:Richard Pinch/sandbox/Ref  | author=I.G. Macdonald | authorlink=Ian G. Macdonald | title=Symmetric Functions and Hall Polynomials | publisher=Oxford University Press | pages=101-104 | year=1979 | isbn=0-19-853530-9 }}
* {{User:Richard Pinch/sandbox/Ref  | author=D.E. Littlewood | title=On certain symmetric functions | journal=Proc. London Math. Soc. | volume=43 | year=1961 | pages=485–498 | doi=10.1112/plms/s3-11.1.485 }}

=Hutchinson operator=
In [[mathematics]], in the study of [[fractal]]s, a '''Hutchinson operator''' is a collection of functions on an underlying space ''E''.  The [[Iterated function|iteration]] on these functions gives rise to an [[iterated function system]], for which the fixed set is [[self-similar]].

==Definition==
Formally, let ''f''<sub>''i''</sub> 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

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

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 ''S''<sub>0</sub> and iterate the actions of the ''f''<sub>''i''</sub>, taking ''S''<sub>''n''+1</sub> to be the union of the images of ''S''<sub>n</sub> under the operator ''H''; then taking ''S'' to be the union of the ''S''<sub>''n''</sub>, that is,

:<math>S_{n+1} = \bigcup_{i=1}^N f_i[S_n] </math>

and

:<math>S = \bigcup_{n=0}^\infty S_n . </math>

==Properties==
Hutchinson (1981) considered the case when the ''f''<sub>''i''</sub> are [[contraction mapping]]s on a Euclidean space ''X'' = '''R'''<sup>d</sup>.  He showed that such a system of functions has a unique [[Compact space|compact]] (closed and bounded) fixed set ''S''.

The collection of functions <math>f_i</math> together with composition form a [[monoid]]. With ''N'' functions, then one may visualize the monoid as a full [[k-ary tree|N-ary tree]] or a [[Cayley tree]].

==References==
* {{cite journal | last=Hutchinson | first=John E. | title=Fractals and self similarity | journal=Indiana Univ. Math. J. | volume=30 | year=1981 | pages=713–747 | doi=10.1512/iumj.1981.30.30055 }}
* {{cite book | title=Chaos and Fractals: New Frontiers of Science | author=Heinz-Otto Peitgen | authorlink=Heinz-Otto Peitgen | coauthors=Hartmut Jürgens, Dietmar Saupe | year=2004 | puclisher=[[Springer-Verlag]] | isbn=0387202293 | pages=84,225 }}

=Kneser theorem=
A statement about [[Sumset|set addition]] in [[finite group]]s.  It may be regarded as an extension of the [[Cauchy-Davenport theorem]] on sumsets in groups of prime order.

Let $G$ be a non-trivial [[abelian group]] and $A,B$ finite non-empty subsets.  If $|A| + |B| \le |G|$ then there is a finite subgroup $H$ of $G$ such that
$$
 |A+B| \ge |A+H| + |B+H| - |H| \ge |A| + |B| - |H| \ .
$$

The subgroup $H$ can be taken to be the ''stabiliser'' of $A+B$
$$
H = \lbrace  g \in G : g + (A+B) = (A+B) \rbrace \ .
$$

==References==
* {{User:Richard Pinch/sandbox/Ref | editor1-last=Geroldinger | editor1-first=Alfred | editor2-last=Ruzsa | editor2-first=Imre Z. | others=Elsholtz, C.; Freiman, G.; Hamidoune, Y. O.; Hegyvári, N.; Károlyi, G.; Nathanson, M.; Solymosi, J.; Stanchescu, Y. With a foreword by Javier Cilleruelo, Marc Noy and Oriol Serra (Coordinators of the DocCourse) | title=Combinatorial number theory and additive group theory | series=Advanced Courses in Mathematics CRM Barcelona | location=Basel | publisher=Birkhäuser | year=2009 | isbn=978-3-7643-8961-1 | zbl=1177.11005 }}
* {{User:Richard Pinch/sandbox/Ref  | first=Martin | last=Kneser | title=Abschätzungen der asymptotischen Dichte von Summenmengen | language=German | journal=Math. Zeitschr. | volume=58 | year=1953 | pages=459–484 | zbl=0051.28104 }}
* {{User:Richard Pinch/sandbox/Ref  | first=Melvyn B. | last=Nathanson | title=Additive Number Theory: Inverse Problems and the Geometry of Sumsets | volume=165 | series=Graduate Texts in Mathematics | publisher=Springer-Verlag | year=1996 | isbn=0-387-94655-1 | zbl=0859.11003 | pages=109–132 }}
* {{User:Richard Pinch/sandbox/Ref  | first1=Terence | last1=Tao | author1-link=Terence Tao | first2=Van H. | last2=Vu | title=Additive Combinatorics | year=2010 | publisher=Cambridge University Press | isbn=978-0-521-13656-3 | zbl=1179.11002 }}

=Linked field=
A [[field]] for which the [[quadratic form]]s attached to [[quaternion algebra]]s have a common property.

Let $F$ be a field of [[Characteristic (algebra)|characteristic]] not equal to 2.  Let $A = (a_1,a_2)_F$, $B = (b_1,b_2)_F$ be quaternion algebras over ''F''.  The algebras $A$ and $B$ are '''linked quaternion algebras''' over $F$ if there is $x \in F$ such that $A$ is equivalent to $(x,y)_F$ and $B$ is equivalent to $(x,z)_F$.

The '''Albert form''' for $A,B$ is
$$
q = \left\langle{ -a_1, -a_2, a_1a_2, b_1, b_2, -b_1b_2 }\right\rangle \ . 
$$
It can be regarded as the difference in the [[Witt ring]] of the ternary forms attached to the imaginary subspaces of $A$ and $B$.  The quaternion algebras are linked if and only if the Albert form is [[isotropic quadratic form|isotropic]].


The field $F$ is ''linked'' if any two quaternion algebras over $F$ are linked.  Every [[global field|global]] and [[local field]] is linked since all quadratic forms of degree 6 over such fields are isotropic.  

The following properties of ''F'' are equivalent:
* $F$ is linked.
* Any two quaternion algebras over ''F'' are linked.
* Every ''Albert form'' (dimension six form of discriminant −1) is isotropic.
* The quaternion algebras form a subgroup of the [[Brauer group]] of ''F''.
* Every dimension five form over ''F'' is a [[Pfister neighbour]].
* No [[biquaternion algebra]] over ''F'' is a [[division algebra]].

A nonreal linked field has [[u-invariant]] equal to 1,2,4 or 8.

==References==
*{{cite book | last=Knus | first=Max-Albert | title=Quadratic and Hermitian forms over rings | series=Grundlehren der Mathematischen Wissenschaften | volume=294 | location=Berlin etc. | publisher=[[Springer-Verlag]] | year=1991 | isbn=3-540-52117-8 | zbl=0756.11008 | page=192 }}
* {{cite book | title=Introduction to Quadratic Forms over Fields | volume=67 | series=Graduate Studies in Mathematics | first=Tsit-Yuen | last=Lam | author-link=T. Y. Lam | publisher=[[American Mathematical Society]] | year=2005 | isbn=0-8218-1095-2 | zbl=1068.11023 | mr = 2104929 }}
* {{cite journal | last=Gentile | first=Enzo R. | title=On linked fields | journal=Rev. Unión Mat. Argent. | volume=35 | year=1990 | pages=67–81 | year=1989 | issn=0041-6932 | zbl=0823.11010 }}

=Pinch point=
A '''pinch point''' or '''cuspidal point''' is a type of [[Singular point of an algebraic variety|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

:<math> f(u,v,w) = u^2 - vw^2 + [4] \, </math>

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

==References==
* {{cite book | author=P. Griffiths | authorlink=Phillip Griffiths | coauthors=[[Joe Harris (mathematician)|J. Harris]] | title=Principles of Algebraic Geometry | series=Wiley Classics Library | publisher=Wiley Interscience | year=1994 | isbn=0-471-05059-8 | page=617 }}

=Quaternionic structure=
''Q-structure''

An axiomatic system that abstracts the concept of a [[quaternion algebra]] over a [[field]].

A ''quaternionic structure'' is a triple $(G,Q,q)$ where $G$ is an [[elementary abelian group]] of [[Exponent of a group|exponent]] 2 with a distinguished element $−1$, $Q$ is a [[pointed set]] with distinguished element $1$, and $q$ is a symmetric [[surjection]] from $G \times G \rightarrow Q$ satisfying axioms
# $q(a,(-1)a) = 1$;
# $q(a,b) = q(a,c) \Leftrightarrow q(q,bc) = 1$;
# $q(a,b) = q(c,d) \Rightarrow \exists x\in Q\,:\, q(a,b) = q(a,x),\, q(c,d) = q(c,x)$.

Every field $F$ supports a  quaternionic structure by taking $G$ to be $F^*/(F^*)^2$, $Q$ the set of [[Brauer class]]es of quaternion algebras in the [[Brauer group]] of $F$ with the [[split quaternion algebra]] as distinguished element and $q(a,b)$ the quaternion algebra $(a,b)_F$.  

==References==
* Tsit-Yuen Lam, ''Introduction to Quadratic Forms over Fields'', Graduate Studies in Mathematics '''67''' , American Mathematical Society (2005) ISBN 0-8218-1095-2 {{ZBL|1068.11023}} {{MR|2104929}}

=Siegel identity=
One of two formulae that are used in the resolution of [[Diophantine equation]]s.

The first formula is
$$
\frac{x_3 - x_1}{x_2 - x_1} + \frac{x_2 - x_3}{x_2 - x_1} = 1 \ .
$$
The second is
$$
\frac{x_3 - x_1}{x_2 - x_1} \cdot\frac{t - x_2}{t - x_3} + \frac{x_2 - x_3}{x_2 - x_1} \cdot \frac{t - x_1}{t - x_3} = 1 \ .
$$

The identities are used in translating Diophantine problems connected with integral points on [[hyperelliptic curve]]s into [[S-unit equation]]s.

See also: [[Siegel formula]].

==References==
* {{cite book | first=Alan | last=Baker | authorlink=Alan Baker (mathematician) | title=Transcendental Number Theory | publisher=[[Cambridge University Press]] | year=1975 | isbn=0-521-20461-5 | zbl=0297.10013 | page=40 }}
* {{cite book | first1=Alan | last1=Baker | authorlink1=Alan Baker (mathematician)| first2=Gisbert | last2= Wüstholz | authorlink2=Gisbert Wüstholz | title=Logarithmic Forms and Diophantine Geometry | series=New Mathematical Monographs | volume=9 | publisher=[[Cambridge University Press]] | year=2007 | isbn=978-0-521-88268-2 | zbl=1145.11004 | page=53 }}
* {{cite book | first1=Daniel S. | last1=Kubert | authorlink1=Daniel Kubert | first2=Serge | last2=Lang | authorlink2=Serge Lang | title=Modular Units | series= Grundlehren der Mathematischen Wissenschaften | volume=244 | year=1981 | isbn=0-387-90517-0 }}
* {{cite book | first=Serge | last=Lang | authorlink=Serge Lang | title=Elliptic Curves: Diophantine Analysis | volume=231 | series=Grundlehren der mathematischen Wissenschaften | publisher=[[Springer-Verlag]] | year=1978 | isbn=0-387-08489-4 }}
* {{cite book | title=The Algorithmic Resolution of Diophantine Equations | volume=41 | series=London Mathematical Society Student Texts | first=N. P. | last=Smart | authorlink=Nigel Smart (cryptographer) | publisher=[[Cambridge University Press]] | year=1998 | isbn=0-521-64633-2 | pages=36–37 }}


=Splicing rule=
A transformation on [[formal language]]s which formalises the action of [[gene splicing]] in [[molecular biology]].  A '''splicing language''' is a language generated by iterated application of a splicing rule: the splicing languages form a proper subset of the [[regular language]]s.

Let $A$ be an alphabet and $L$ a language over $A$, that is, a subset of the [[free monoid]] $A^*$.  A '''splicing rule''' is a quadruple $r = (a,b,c,d)$ of elements of $A^*$, and the action of the rule $r$ on $L$ is to produce the language
$$
r(L) = \{ xady : xabq, pcdy \in L \} \ . 
$$
If $R$ is a set of rules then $R(L)$ is the union of the languages produced by the rules of $R$.  We say that $R$ ''respects'' $L$ if $R(L)$ is a subset of $L$.  The $R$-closure of $L$ is the union of $L$ and all iterates of $R$ on$L$: clearly it is respected by $R$.  A '''splicing language''' is the $R$-closure of a finite language.<ref name=A236>Anderson (2006) p.236</ref>

A rule set $R$ is '''reflexive''' if $(a,b,c,d)$ in $R$ implies that $(a,b,a,b)$ and $(c,d,c,d)$ are in $R$.  A splicing language is reflexive if it is defined by a reflexive rule set.<ref name=A242/>

Let $A = \{a,b,c\}$.  The rule $(caba,a,cab,a)$ applied to the finite set $\{cabb,cabab,cabaab\}$ generates the regular language $caba^*b$.<ref name=A238>Anderson (2006) p.238</ref>

==Properties==
* All splicing languages are regular.<ref name=A239>Anderson (2006) p.239</ref>
* Not all regular languages are splicing.<ref name=A240>Anderson (2006) p.240</ref>  An example is $(aa)^*$ over $\{a,b\}$.<ref name=A239/>  
* If $L$ is a regular language on the alphabet $A$, and $z$ is a letter not in $A$, then the language $\{ zw : w \in L \}$ is a splicing language.<ref name=A238/>
* There is an algorithm to determine whether a given regular language is a reflexive splicing language.<ref name=A242>Anderson (2006) p.242</ref>
* The set of splicing rules that respect a regular language can be determined from the [[syntactic monoid]] of the language.<ref name=A241>Anderson (2006) p.241</ref>

== References ==
{{reflist}}
* {{cite book | last=Anderson | first=James A. | title=Automata theory with modern applications | others=With contributions by Tom Head | location=Cambridge | publisher=[[Cambridge University Press]] | year=2006 | isbn=0-521-61324-8 | zbl=1127.68049 }}

=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 part]]s
$$
\left\lbrace x (3/2)^ n \right\rbrace
$$
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==
* {{cite book | last1=Everest | first1=Graham | last2=van der Poorten | first2=Alf | last3=Shparlinski | first3=Igor | last4=Ward | first4=Thomas | title=Recurrence sequences | series=Mathematical Surveys and Monographs | volume=104 | location=[[Providence, RI]] | publisher=[[American Mathematical Society]] | year=2003 | isbn=0-8218-3387-1 | zbl=1033.11006 }}