Schanuel's conjecture
Encyclopedia
In mathematics
, specifically transcendence theory
, Schanuel's conjecture is a conjecture made by Stephen Schanuel
in the 1960s concerning the transcendence degree
of certain field extension
s of the rational numbers.
The conjecture can be found in Lang (1966). No proof is known.
. The special case where the numbers z1,...,zn are all algebraic
is the Lindemann–Weierstrass theorem
. If, on the other hand, the numbers are chosen so as to make exp(z1),...,exp(zn) all algebraic then one would prove that linearly independent logarithms of algebraic numbers are algebraically independent, a strengthening of Baker's theorem
.
The Gelfond–Schneider theorem
follows from this strengthened version of Baker's theorem, as does the currently unproven four exponentials conjecture
.
Schanuel's conjecture, if proved, would also settle the algebraic nature of numbers such as e + π and ee, and prove that e and π are algebraically independent simply by setting z1 = 1 and z2 = πi, and using Euler's identity.
Euler's identity states that eπi + 1 = 0. If Schanuel's conjecture is true then this is, in some precise sense involving exponential rings, the only relation between e, π, and i over the complex numbers.
Although ostensibly a problem in number theory, the conjecture has implications in model theory
as well. Angus Macintyre
and Alex Wilkie
, for example, proved that the theory of the real field with exponentiation, Rexp, is decidable
provided Schanuel's conjecture is true. In fact they only needed the real version of the conjecture, defined below, to prove this result, which would be a positive solution to Tarski's exponential function problem
.
A version of Schanuel's conjecture for formal power series
, also by Schanuel, was proven by James Ax
in 1971. It states:
As stated above, the decidability of Rexp follows from the real version of Schanuel's conjecture which is as follows:
A related conjecture called the uniform real Schanuel's conjecture essentially says the same but puts a bound on the integers mi. The uniform real version of the conjecture is equivalent to the standard real version. Macintyre and Wilkie showed that a consequence of Schanuel's conjecture, which they dubbed the Weak Schanuel's conjecture, was equivalent to the decidability of Rexp. This conjecture states that there is a computable upper bound on the norm of non-singular solutions to systems of exponential polynomial
s, a fact that is not an immediately obvious consequence of Schanuel's conjecture.
It is also known that Schanuel's conjecture would be a consequence of conjectural results in the theory of motives
. There Grothendieck's period conjecture for an abelian variety
A states that the transcendence degree of its period matrix is the same as the dimension of the associated Mumford–Tate group
, and what is known by work of Pierre Deligne
is that the dimension is an upper bound for the transcendence degree. Bertolin has shown how a generalised period conjecture includes Schanuel's conjecture.
In 2004, Boris Zilber systematically constructed exponential field
s Kexp that were algebraically closed and of characteristic zero, and such that one of these fields existed for each uncountable cardinality. He axiomatised these fields and, using Hrushovski's construction
, he proved that for each uncountable cardinal this theory of "pseudo-exponentiation" is satisfiable, and categorical for a particular field, this field he called the canonical one for that cardinality. The canonical field for the cardinality of the continuum satisfies Schanuel's conjecture, and so if in this case Kexp = Cexp then Schanuel's conjecture is true. Unfortunately one of the criteria for this model to be the complex numbers with exponentiation is that Schanuel's conjecture is true, as well as another unproven property of the complex numbers with exponentiation, which Zilber calls exponential-algebraic closedness.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, specifically transcendence theory
Transcendence theory
Transcendence theory is a branch of number theory that investigates transcendental numbers, in both qualitative and quantitative ways.-Transcendence:...
, Schanuel's conjecture is a conjecture made by Stephen Schanuel
Stephen Schanuel
Stephen H. Schanuel is an American mathematician working in the fields of abstract algebra and number theory, more specifically category theory and measure theory....
in the 1960s concerning the transcendence degree
Transcendence degree
In abstract algebra, the transcendence degree of a field extension L /K is a certain rather coarse measure of the "size" of the extension...
of certain field extension
Field extension
In abstract algebra, field extensions are the main object of study in field theory. The general idea is to start with a base field and construct in some manner a larger field which contains the base field and satisfies additional properties...
s of the rational numbers.
Statement
The conjecture is as follows:- Given any n complex numberComplex numberA complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
s z1,...,zn which are linearly independentLinear independenceIn linear algebra, a family of vectors is linearly independent if none of them can be written as a linear combination of finitely many other vectors in the collection. A family of vectors which is not linearly independent is called linearly dependent...
over the rational numberRational numberIn mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...
s Q, the extension fieldField extensionIn abstract algebra, field extensions are the main object of study in field theory. The general idea is to start with a base field and construct in some manner a larger field which contains the base field and satisfies additional properties...
Q(z1,...,zn, expExponential functionIn mathematics, the exponential function is the function ex, where e is the number such that the function ex is its own derivative. The exponential function is used to model a relationship in which a constant change in the independent variable gives the same proportional change In mathematics,...
(z1),...,exp(zn)) has transcendence degreeTranscendence degreeIn abstract algebra, the transcendence degree of a field extension L /K is a certain rather coarse measure of the "size" of the extension...
of at least n over Q.
The conjecture can be found in Lang (1966). No proof is known.
Consequences
The conjecture, if proven, would subsume most known results in transcendental number theoryTranscendence theory
Transcendence theory is a branch of number theory that investigates transcendental numbers, in both qualitative and quantitative ways.-Transcendence:...
. The special case where the numbers z1,...,zn are all algebraic
Algebraic number
In mathematics, an algebraic number is a number that is a root of a non-zero polynomial in one variable with rational coefficients. Numbers such as π that are not algebraic are said to be transcendental; almost all real numbers are transcendental...
is the Lindemann–Weierstrass theorem
Lindemann–Weierstrass theorem
In mathematics, the Lindemann–Weierstrass theorem is a result that is very useful in establishing the transcendence of numbers. It states that if 1, ..., are algebraic numbers which are linearly independent over the rational numbers ', then 1, ..., are algebraically...
. If, on the other hand, the numbers are chosen so as to make exp(z1),...,exp(zn) all algebraic then one would prove that linearly independent logarithms of algebraic numbers are algebraically independent, a strengthening of Baker's theorem
Baker's theorem
In transcendence theory, a mathematical discipline, Baker's theorem gives a lower bound for linear combinations of logarithms of algebraic numbers...
.
The Gelfond–Schneider theorem
Gelfond–Schneider theorem
In mathematics, the Gelfond–Schneider theorem establishes the transcendence of a large class of numbers. It was originally proved independently in 1934 by Aleksandr Gelfond and Theodor Schneider...
follows from this strengthened version of Baker's theorem, as does the currently unproven four exponentials conjecture
Four exponentials conjecture
In mathematics, specifically the field of transcendental number theory, the four exponentials conjecture is a conjecture which, given the right conditions on the exponents, would guarantee the transcendence of at least one of four exponentials...
.
Schanuel's conjecture, if proved, would also settle the algebraic nature of numbers such as e + π and ee, and prove that e and π are algebraically independent simply by setting z1 = 1 and z2 = πi, and using Euler's identity.
Euler's identity states that eπi + 1 = 0. If Schanuel's conjecture is true then this is, in some precise sense involving exponential rings, the only relation between e, π, and i over the complex numbers.
Although ostensibly a problem in number theory, the conjecture has implications in model theory
Model theory
In mathematics, model theory is the study of mathematical structures using tools from mathematical logic....
as well. Angus Macintyre
Angus MacIntyre
Angus John Macintyre is a British mathematician known for his contributions to Model theory and logic. He is professor of mathematics at the Queen Mary, University of London....
and Alex Wilkie
Alex Wilkie
Alex Wilkie FRS is a British mathematician known for his contributions to Model theory and logic. Previously Reader in Mathematical Logic at the University of Oxford, he was appointed to the Fielden Chair of Pure Mathematics at the University of Manchester in 2007.Wilkie attended Aylesbury...
, for example, proved that the theory of the real field with exponentiation, Rexp, is decidable
Decidability (logic)
In logic, the term decidable refers to the decision problem, the question of the existence of an effective method for determining membership in a set of formulas. Logical systems such as propositional logic are decidable if membership in their set of logically valid formulas can be effectively...
provided Schanuel's conjecture is true. In fact they only needed the real version of the conjecture, defined below, to prove this result, which would be a positive solution to Tarski's exponential function problem
Tarski's exponential function problem
In model theory, Tarski's exponential function problem asks whether the usual theory of the real numbers together with the exponential function is decidable...
.
Related conjectures and results
The converse Schanuel conjecture is the following statement:- Suppose F is a countable fieldField (mathematics)In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
with characteristicCharacteristic (algebra)In mathematics, the characteristic of a ring R, often denoted char, is defined to be the smallest number of times one must use the ring's multiplicative identity element in a sum to get the additive identity element ; the ring is said to have characteristic zero if this repeated sum never reaches...
0, and e : F → F is a homomorphismGroup homomorphismIn mathematics, given two groups and , a group homomorphism from to is a function h : G → H such that for all u and v in G it holds that h = h \cdot h...
from the additive group (F,+) to the multiplicative group (F,·) whose kernelKernel (algebra)In the various branches of mathematics that fall under the heading of abstract algebra, the kernel of a homomorphism measures the degree to which the homomorphism fails to be injective. An important special case is the kernel of a matrix, also called the null space.The definition of kernel takes...
is cyclicCyclic groupIn group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element g such that, when written multiplicatively, every element of the group is a power of g .-Definition:A group G is called cyclic if there exists an element g...
. Suppose further that for any n elements x1,...,xn of F which are linearly independent over Q, the extension field Q(x1,...,xn,e(x1),...,e(xn)) has transcendence degree at least n over Q. Then there exists a field homomorphism h : F → C such that h(e(x))=exp(h(x)) for all x in F.
A version of Schanuel's conjecture for formal power series
Formal power series
In mathematics, formal power series are a generalization of polynomials as formal objects, where the number of terms is allowed to be infinite; this implies giving up the possibility to substitute arbitrary values for indeterminates...
, also by Schanuel, was proven by James Ax
James Ax
James Burton Ax was a mathematician who proved several results in algebra and number theory by using model theory. He shared the seventh Frank Nelson Cole Prize in Number Theory with Simon B. Kochen, which was awarded for a series of three joint papers on Diophantine problems.James Ax earned his...
in 1971. It states:
- Given any n formal power series f1,...,fn in tC
t which are linearly independent over Q, then the field extension C(t,f1,...,fn,exp(f1),...,exp(fn)) has transcendence degree at least n over C(t).
As stated above, the decidability of Rexp follows from the real version of Schanuel's conjecture which is as follows:
- Suppose x1,...,xn are real numberReal numberIn mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
s and the transcendence degree of the field Q(x1,...,xn, expExponential functionIn mathematics, the exponential function is the function ex, where e is the number such that the function ex is its own derivative. The exponential function is used to model a relationship in which a constant change in the independent variable gives the same proportional change In mathematics,...
(x1),...,exp(xn)) is strictly less than n, then there are integers m1,...,mn, not all zero, such that m1x1 +...+ mnxn = 0.
A related conjecture called the uniform real Schanuel's conjecture essentially says the same but puts a bound on the integers mi. The uniform real version of the conjecture is equivalent to the standard real version. Macintyre and Wilkie showed that a consequence of Schanuel's conjecture, which they dubbed the Weak Schanuel's conjecture, was equivalent to the decidability of Rexp. This conjecture states that there is a computable upper bound on the norm of non-singular solutions to systems of exponential polynomial
Exponential polynomial
In mathematics, exponential polynomials are functions on fields, rings, or abelian groups that take the form of polynomials in a variable and an exponential function.-In fields:...
s, a fact that is not an immediately obvious consequence of Schanuel's conjecture.
It is also known that Schanuel's conjecture would be a consequence of conjectural results in the theory of motives
Motive (algebraic geometry)
In algebraic geometry, a motive denotes 'some essential part of an algebraic variety'. To date, pure motives have been defined, while conjectural mixed motives have not. Pure motives are triples , where X is a smooth projective variety, p : X ⊢ X is an idempotent correspondence, and m an integer...
. There Grothendieck's period conjecture for an abelian variety
Abelian variety
In mathematics, particularly in algebraic geometry, complex analysis and number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular functions...
A states that the transcendence degree of its period matrix is the same as the dimension of the associated Mumford–Tate group
Mumford–Tate group
In algebraic geometry, the Mumford–Tate group MT constructed from a Hodge structure F is a certain algebraic group G, named for David Mumford and John Tate. When F is given by a rational representation of an algebraic torus, the definition of G is as the Zariski closure of the image in the...
, and what is known by work of Pierre Deligne
Pierre Deligne
- See also :* Deligne conjecture* Deligne–Mumford moduli space of curves* Deligne–Mumford stacks* Deligne cohomology* Fourier–Deligne transform* Langlands–Deligne local constant- External links :...
is that the dimension is an upper bound for the transcendence degree. Bertolin has shown how a generalised period conjecture includes Schanuel's conjecture.
Zilber's pseudo-exponentiation
While a proof of Schanuel's conjecture with number theoretic tools seems a long way off, an approach via model theory has prompted a surge of research on the conjecture.In 2004, Boris Zilber systematically constructed exponential field
Exponential field
In mathematics, an exponential field is a field that has an extra operation on its elements which extends the usual idea of exponentiation.-Definition:...
s Kexp that were algebraically closed and of characteristic zero, and such that one of these fields existed for each uncountable cardinality. He axiomatised these fields and, using Hrushovski's construction
Hrushovski construction
In model theory, a branch of mathematical logic, the Hrushovski construction generalizes the Fraïssé limit by working with a notion of strong substructure \leq rather than \subseteq. It can be thought of as a kind of "model-theoretic forcing", where a stable structure is created, called the generic...
, he proved that for each uncountable cardinal this theory of "pseudo-exponentiation" is satisfiable, and categorical for a particular field, this field he called the canonical one for that cardinality. The canonical field for the cardinality of the continuum satisfies Schanuel's conjecture, and so if in this case Kexp = Cexp then Schanuel's conjecture is true. Unfortunately one of the criteria for this model to be the complex numbers with exponentiation is that Schanuel's conjecture is true, as well as another unproven property of the complex numbers with exponentiation, which Zilber calls exponential-algebraic closedness.