Transcendental number theory is a branch ofnumber theory that investigatestranscendental numbers (numbers that are not solutions of anypolynomial equation withrationalcoefficients), in both qualitative and quantitative ways.

Thefundamental theorem of algebra tells us that if we have a non-constantpolynomial with rational coefficients (or equivalently, byclearing denominators, withinteger coefficients) then that polynomial will have aroot in thecomplex numbers. That is, for any non-constant polynomial${\displaystyle P}$ with rational coefficients there will be a complex number${\displaystyle \alpha }$ such that${\displaystyle P(\alpha )=0}$. Transcendence theory is concerned with the converse question: given a complex number${\displaystyle \alpha }$, is there a polynomial${\displaystyle P}$ with rational coefficients such that${\displaystyle P(\alpha )=0?}$ If no such polynomial exists then the number is called transcendental.

More generally the theory deals withalgebraic independence of numbers. A set of numbers {α₁, α₂, …, α_n} is called algebraically independent over afieldK if there is no non-zero polynomialP inn variables with coefficients inK such thatP(α₁, α₂, …, α_n) = 0. So working out if a given number is transcendental is really a special case of algebraic independence wheren = 1 and the fieldK is the field ofrational numbers.

A related notion is whether there is aclosed-form expression for a number, including exponentials and logarithms as well as algebraic operations. There are various definitions of "closed-form", and questions about closed-form can often be reduced to questions about transcendence.

Use of the termtranscendental to refer to an object that is not algebraic dates back to the seventeenth century, whenGottfried Leibniz proved that thesine function was not analgebraic function.^[1] The question of whether certain classes of numbers could be transcendental dates back to 1748^[2] whenEuler asserted^[3] that the number log_ab was notalgebraic forrational numbersa andb providedb is not of the formb = a^c for some rationalc.

Euler's assertion was not proved until the twentieth century, but almost a hundred years after his claimJoseph Liouville did manage to prove the existence of numbers that are not algebraic, something that until then had not been known for sure.^[4] His original papers on the matter in the 1840s sketched out arguments usingsimple continued fractions to construct transcendental numbers. Later, in the 1850s, he gave anecessary condition for a number to be algebraic, and thus a sufficient condition for a number to be transcendental.^[5] This transcendence criterion was not strong enough to be necessary too, and indeed it fails to detect that the numbere is transcendental. But his work did provide a larger class of transcendental numbers, now known asLiouville numbers in his honour.

Liouville's criterion essentially said that algebraic numbers cannot be very well approximated by rational numbers. So if a number can be very well approximated by rational numbers then it must be transcendental. The exact meaning of "very well approximated" in Liouville's work relates to a certain exponent. He showed that if α is analgebraic number of degreed ≥ 2 and ε is any number greater than zero, then the expression

can be satisfied by only finitely many rational numbersp/q. Using this as a criterion for transcendence is not trivial, as one must check whether there are infinitely many solutionsp/q for everyd ≥ 2.

In the twentieth century work byAxel Thue,^[6]Carl Siegel,^[7] andKlaus Roth^[8] reduced the exponent in Liouville's work fromd + ε tod/2 + 1 + ε, and finally, in 1955, to 2 + ε. This result, known as theThue–Siegel–Roth theorem, is ostensibly the best possible, since if the exponent 2 + ε is replaced by just 2 then the result is no longer true. However,Serge Lang conjectured an improvement of Roth's result; in particular he conjectured thatq^2+ε in the denominator of the right-hand side could be reduced to${\displaystyle q^2(\log q{)}^{1+ϵ}}$.

Roth's work effectively ended the work started by Liouville, and his theorem allowed mathematicians to prove the transcendence of many more numbers, such as theChampernowne constant. The theorem is still not strong enough to detectall transcendental numbers, though, and many famous constants includinge and π either are not or are not known to be very well approximable in the above sense.^[9]

Fortunately other methods were pioneered in the nineteenth century to deal with the algebraic properties ofe, and consequently of π throughEuler's identity. This work centred on use of the so-calledauxiliary function. These arefunctions which typically have many zeros at the points under consideration. Here "many zeros" may mean many distinct zeros, or as few as one zero but with a highmultiplicity, or even many zeros all with high multiplicity.Charles Hermite used auxiliary functions that approximated the functions${\displaystyle e^{kx}}$ for eachnatural number${\displaystyle k}$ in order to prove the transcendence of${\displaystyle e}$ in 1873.^[10] His work was built upon byFerdinand von Lindemann in the 1880s^[11] in order to prove thate^α is transcendental for nonzero algebraic numbers α. In particular this proved that π is transcendental sincee^πi is algebraic, and thus answered in the negative theproblem of antiquity as to whether it was possible tosquare the circle.Karl Weierstrass developed their work yet further and eventually proved theLindemann–Weierstrass theorem in 1885.^[12]

In 1900David Hilbert posed his famouscollection of problems. Theseventh of these, and one of the hardest in Hilbert's estimation, asked about the transcendence of numbers of the forma^b wherea andb are algebraic,a is not zero or one, andb isirrational. In the 1930sAlexander Gelfond^[13] andTheodor Schneider^[14] proved that all such numbers were indeed transcendental using a non-explicit auxiliary function whose existence was granted bySiegel's lemma. This result, theGelfond–Schneider theorem, proved the transcendence of numbers such ase^π and theGelfond–Schneider constant.

The next big result in this field occurred in the 1960s, whenAlan Baker made progress on a problem posed by Gelfond onlinear forms in logarithms. Gelfond himself had managed to find a non-trivial lower bound for the quantity

${\displaystyle |{\beta }_1\log {\alpha }_1+{\beta }_2\log {\alpha }_2|}$

where all four unknowns are algebraic, the αs being neither zero nor one and the βs being irrational. Finding similar lower bounds for the sum of three or more logarithms had eluded Gelfond, though. The proof ofBaker's theorem contained such bounds, solving Gauss'class number problem for class number one in the process. This work won Baker theFields Medal for its uses in solvingDiophantine equations. From a purely transcendental number theoretic viewpoint, Baker had proved that if α₁, ..., α_n are algebraic numbers, none of them zero or one, and β₁, ..., β_n are algebraic numbers such that 1, β₁, ..., β_n arelinearly independent over the rational numbers, then the number

${\displaystyle {\alpha }_1^{{\beta }_1}{\alpha }_2^{{\beta }_2}\cdots {\alpha }_n^{{\beta }_n}}$

is transcendental.^[15]

In the 1870s,Georg Cantor started to developset theory and, in 1874, published apaper proving that the algebraic numbers could be put inone-to-one correspondence with the set ofnatural numbers, and thus that the set of transcendental numbers must beuncountable.^[16] Later, in 1891, Cantor used his more familiardiagonal argument to prove the same result.^[17] While Cantor's result is often quoted as being purely existential and thus unusable for constructing a single transcendental number,^[18][19] the proofs in both the aforementioned papers give methods to construct transcendental numbers.^[20]

While Cantor used set theory to prove the plenitude of transcendental numbers, a recent development has been the use ofmodel theory in attempts to prove anunsolved problem in transcendental number theory. The problem is to determine thetranscendence degree of the field

${\displaystyle K=\mathbb{Q}(x_1,\dots ,x_n,e^{x_1},\dots ,e^{x_n})}$

for complex numbersx₁, ...,x_n that are linearly independent over the rational numbers.Stephen Schanuelconjectured that the answer is at leastn, but no proof is known. In 2004, though,Boris Zilber published a paper that used model theoretic techniques to create a structure that behaves very much like thecomplex numbers equipped with the operations of addition, multiplication, and exponentiation. Moreover, in this abstract structure Schanuel's conjecture does indeed hold.^[21] Unfortunately it is not yet known that this structure is in fact the same as the complex numbers with the operations mentioned; there could exist some other abstract structure that behaves very similarly to the complex numbers but where Schanuel's conjecture doesn't hold. Zilber did provide several criteria that would prove the structure in question wasC, but could not prove the so-called Strong Exponential Closure axiom. The simplest case of this axiom has since been proved,^[22] but a proof that it holds in full generality is required to complete the proof of the conjecture.

A typical problem in this area of mathematics is to work out whether a given number is transcendental.Cantor used acardinality argument to show that there are onlycountably many algebraic numbers, and hencealmost all numbers are transcendental. Transcendental numbers therefore represent the typical case; even so, it may be extremely difficult to prove that a given number is transcendental (or even simply irrational).

For this reason transcendence theory often works towards a more quantitative approach. So given a particular complex number α one can ask how close α is to being an algebraic number. For example, if one supposes that the number α is algebraic then can one show that it must have very high degree or a minimum polynomial with very large coefficients? Ultimately if it is possible to show that no finite degree or size of coefficient is sufficient then the number must be transcendental. Since a number α is transcendental if and only ifP(α) ≠ 0 for every non-zero polynomialP with integer coefficients, this problem can be approached by trying to find lower bounds of the form

${\displaystyle |P(a)|>F(A,d)}$

where the right hand side is some positive function depending on some measureA of the size of thecoefficients ofP, and itsdegreed, and such that these lower bounds apply to allP ≠ 0. Such a bound is called atranscendence measure.

The case ofd = 1 is that of "classical"diophantine approximation asking for lower bounds for

${\displaystyle |ax+b|}$.

The methods of transcendence theory and diophantine approximation have much in common: they both use theauxiliary function concept.

TheGelfond–Schneider theorem was the major advance in transcendence theory in the period 1900–1950. In the 1960s the method ofAlan Baker onlinear forms in logarithms ofalgebraic numbers reanimated transcendence theory, with applications to numerous classical problems anddiophantine equations.

Kurt Mahler in 1932 partitioned the transcendental numbers into 3 classes, calledS,T, andU.^[23] Definition of these classes draws on an extension of the idea of aLiouville number (cited above).

One way to define a Liouville number is to consider how small a givenreal numberx makes linear polynomials |qx − p| without making them exactly 0. Herep,q are integers with |p|, |q| bounded by a positive integer H.

Let${\displaystyle m(x,1,H)}$ be the minimum non-zero absolute value these polynomials take and take:

${\displaystyle \omega (x,1,H)=-\frac{\log m(x,1,H)}{\log H}}$

${\displaystyle \omega (x,1)=\underset{H\to \mathrm{\infty }}{lim sup}\omega (x,1,H).}$

ω(x, 1) is often called themeasure of irrationality of a real number x. For rational numbers, ω(x, 1) = 0 and is at least 1 for irrational real numbers. A Liouville number is defined to have infinite measure of irrationality.Roth's theorem says that irrational real algebraic numbers have measure of irrationality 1.

Next consider the values of polynomials at a complex numberx, when these polynomials have integer coefficients, degree at mostn, andheight at mostH, withn,H being positive integers.

Let${\displaystyle m(x,n,H)}$ be the minimum non-zero absolute value such polynomials take at${\displaystyle x}$ and take:

${\displaystyle \omega (x,n,H)=-\frac{\log m(x,n,H)}{n\log H}}$

${\displaystyle \omega (x,n)=\underset{H\to \mathrm{\infty }}{lim sup}\omega (x,n,H).}$

Suppose this is infinite for some minimum positive integer n. A complex numberx in this case is called aU number of degree n.

Now we can define

${\displaystyle \omega (x)=\underset{n\to \mathrm{\infty }}{lim sup}\omega (x,n).}$

ω(x) is often called themeasure of transcendence of x. If the ω(x,n) are bounded, then ω(x) is finite, andx is called anS number. If the ω(x,n) are finite but unbounded,x is called aT number.x is algebraic if and only if ω(x) = 0.

Clearly the Liouville numbers are a subset of the U numbers.William LeVeque in 1953 constructed U numbers of any desired degree.^[24] TheLiouville numbers and hence the U numbers are uncountable sets. They are sets of measure 0.^[25]

T numbers also comprise a set of measure 0.^[26] It took about 35 years to show their existence.Wolfgang M. Schmidt in 1968 showed that examples exist. However,almost all complex numbers are S numbers.^[27] Mahler proved that the exponential function sends all non-zero algebraic numbers to S numbers:^[28][29] this shows thate is an S number and gives a proof of the transcendence ofπ. This numberπ is known not to be a U number.^[30] Many other transcendental numbers remain unclassified.

Two numbersx,y are calledalgebraically dependent if there is a non-zero polynomialP in two indeterminates with integer coefficients such thatP(x, y) = 0. There is a powerful theorem that two complex numbers that are algebraically dependent belong to the same Mahler class.^[24][31] This allows construction of new transcendental numbers, such as the sum of a Liouville number withe or π.

The symbol S probably stood for the name of Mahler's teacherCarl Ludwig Siegel, and T and U are just the next two letters.

Jurjen Koksma in 1939 proposed another classification based on approximation by algebraic numbers.^[23][32]

Consider the approximation of a complex numberx by algebraic numbers of degree ≤ n and height ≤ H. Let α be an algebraic number of this finite set such that |x − α| has the minimum positive value. Define ω*(x,H,n) and ω*(x,n) by:

${\displaystyle |x-\alpha |=H^{-n{\omega }^{\ast }(x,H,n)-1}.}$

${\displaystyle {\omega }^{\ast }(x,n)=\underset{H\to \mathrm{\infty }}{lim sup}{\omega }^{\ast }(x,n,H).}$

If for a smallest positive integern, ω*(x,n) is infinite,x is called aU*-number of degree n.

If the ω*(x,n) are bounded and do not converge to 0,x is called anS*-number,

A numberx is called anA*-number if the ω*(x,n) converge to 0.

If the ω*(x,n) are all finite but unbounded,x is called aT*-number,

Koksma's and Mahler's classifications are equivalent in that they divide the transcendental numbers into the same classes.^[32] TheA*-numbers are the algebraic numbers.^[27]

Let

${\displaystyle \lambda =\frac{1}{3}+\sum _{k=1}^{\mathrm{\infty }}{10}^{-k!}.}$

It can be shown that thenth root of λ (a Liouville number) is a U-number of degreen.^[33]

This construction can be improved to create an uncountable family of U-numbers of degreen. LetZ be the set consisting of every other power of 10 in the series above for λ. The set of all subsets ofZ is uncountable. Deleting any of the subsets ofZ from the series for λ creates uncountably many distinct Liouville numbers, whosenth roots are U-numbers of degreen.

Thesupremum of the sequence {ω(x, n)} is called thetype. Almost all real numbers are S numbers of type 1, which is minimal for real S numbers. Almost all complex numbers are S numbers of type 1/2, which is also minimal. The claims of almost all numbers were conjectured by Mahler and in 1965 proved by Vladimir Sprindzhuk.^[34]

While the Gelfond–Schneider theorem proved that a large class of numbers was transcendental, this class was still countable. Many well-known mathematical constants are still not known to be transcendental, and in some cases it is not even known whether they are rational or irrational. A partial list can be foundhere.

A major problem in transcendence theory is showing that a particular set of numbers is algebraically independent rather than just showing that individual elements are transcendental. So while we know thate andπ are transcendental that doesn't imply thate + π is transcendental, nor other combinations of the two (excepte^π,Gelfond's constant, which is known to be transcendental). Another major problem is dealing with numbers that are not related to the exponential function. The main results in transcendence theory tend to revolve arounde and the logarithm function, which means that wholly new methods tend to be required to deal with numbers that cannot be expressed in terms of these two objects in an elementary fashion.

Schanuel's conjecture would solve the first of these problems somewhat as it deals with algebraic independence and would indeed confirm thate +π is transcendental. It still revolves around the exponential function, however, and so would not necessarily deal with numbers such asApéry's constant or theEuler–Mascheroni constant. Another extremely difficult unsolved problem is the so-calledconstant or identity problem.^[35]

• Baker, Alan (1975).Transcendental Number Theory. paperback edition 1990.Cambridge University Press.ISBN 0-521-20461-5.Zbl 0297.10013.
• Bugeaud, Yann (2012).Distribution modulo one and Diophantine approximation. Cambridge Tracts in Mathematics. Vol. 193.Cambridge University Press.ISBN 978-0-521-11169-0.Zbl 1260.11001.
• Burger, Edward B.; Tubbs, Robert (2004).Making transcendence transparent. An intuitive approach to classical transcendental number theory.Springer.ISBN 978-0-387-21444-3.Zbl 1092.11031.
• Gelfond, A. O. (1960).Transcendental and Algebraic Numbers. Dover.Zbl 0090.26103.
• Lang, Serge (1966).Introduction to Transcendental Numbers. Addison–Wesley.Zbl 0144.04101.
• LeVeque, William J. (2002) [1956].Topics in Number Theory, Volumes I and II. Dover.ISBN 978-0-486-42539-9.
• Natarajan, Saradha[in French]; Thangadurai, Ravindranathan (2020).Pillars of Transcendental Number Theory.Springer Verlag.ISBN 978-981-15-4154-4.
• Sprindzhuk, Vladimir G. (1969).Mahler's Problem in Metric Number Theory (1967). AMS Translations of Mathematical Monographs. Translated from Russian by B. Volkmann.American Mathematical Society.ISBN 978-1-4704-4442-6.
• Sprindzhuk, Vladimir G. (1979).Metric theory of Diophantine approximations. Scripta Series in Mathematics. Translated from Russian by Richard A. Silverman. Foreword by Donald J. Newman.Wiley.ISBN 0-470-26706-2.Zbl 0482.10047.

• Alan Baker andGisbert Wüstholz,Logarithmic Forms and Diophantine Geometry, New Mathematical Monographs9, Cambridge University Press, 2007,ISBN 978-0-521-88268-2

• Analytic number theory
• Transcendental numbers

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