Transcendence theory

In mathematics, transcendence theory is a branch of number theory that investigates transcendental numbers, in both qualitative and quantitative ways.

Transcendence

The fundamental theorem of algebra tells us that if we have a non-zero polynomial with integer coefficients then that polynomial will have a root in the complex numbers. That is, for any polynomial "P" with integer coefficients there will be a complex number α such that "P"(α) = 0. Transcendence theory is concerned with the converse question, given a complex number α, is there a polynomial "P" with integer coefficients such that "P"(α) = 0? If no such polynomial exists then the number is called transcendental.

More generally the theory deals with algebraic independence of numbers. A set of numbers {α₁,α₂,…,α_n} is called algebraically independent over a field "k" if there is no non-zero polynomial "P" in "n" variables with coefficients in "k" such that "P"(α₁,α₂,…,α_n) = 0. So working out if a given number is transcendental is really a special case of algebraic independence where our set consists of just one number.

History

Approximation by rational numbers: Liouville to Roth

Use of the term "transcendental" to refer to an object that is not algebraic dates back to the seventeenth century, when Gottfried Leibniz proved that the sine function was not an algebraic function [N. Bourbaki, "Elements of the History of Mathematics" Springer (1994).] . The question of whether certain classes of numbers could be transcendental dates back to 1748 [A.O. Gelfond, "Transcendental and algebraic numbers", Dover Publications Inc., (1960), p.2.] when Euler asserted [L. Euler, "Introductio in analysin infinitorum", Lausanne, 1748.] that the number log_"a""b" was not algebraic for rational numbers "a" and "b" provided "b" is not of the form "b" = "a"^"c" for some algebraic number "c".

Euler's assertion was not proved until the twentieth century, but almost a hundred years after his claim Joseph Liouville did manage to prove the existence of numbers that are not algebraic, something that until then had not been known for sure. His original papers on the matter in the 1840s sketched out arguments using continued fractions to construct transcendental numbers. Later, in the 1850s, he gave a necessary condition for a number to be algebraic, and thus a sufficient condition for a number to be transcendental [J. Liouville, "Sur les classes très étendues de quantités dont la valeur n'est ni algébrique ni même réductible à des irrationelles algébriques", Comptes Rendus Acad. Sci. Paris 18, (1844), pp.883–885, 910–911; Journal Math. Pures et Appl. 16, (1851), pp.133–142.] . This transcendence criterion was not strong enough to be necessary too, and indeed it fails to detect that the number "e" is transcendental. But his work did provide a larger class of transcendental numbers, now known as Liouville 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 an algebraic number of degree "d" ≥ 2 and ε is any number greater than zero, then the expression:can be satisfied by only finitely many rational numbers "p"/"q". Using this as a criterion for transcendence is not trivial, as one must check there are infinitely many solutions "p"/"q" for every "d" ≥ 2.

In the twentieth century work by Axel Thue [A. Thue, "Über Annäherungswerte algebraischer Zahlen", Journal f.d. reine u. angew. Math. 135, (1909), pp.284–305.] , Carl Siegel [C.L. Siegel, "Approximation algebraischer Zahlen", Math. Zeitschrift 10, (1921), pp.172–213.] , and Klaus Roth [ K.F. Roth, "Rational approximations to algebraic numbers" and "Corrigendum", Mathematika 2, (1955), pp.1–20; p.168.] reduced the exponent in Liouville's work from "d" + ε to "d"/2 + 1 + ε, and finally, in 1955, to 2 + ε. This result, known as the Thue-Siegel-Roth theorem, is best possible, since if the exponent 2 + ε is replaced by just 2 then the result is no longer true.

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 the Champernowne constant. The theorem is still not strong enough to detect "all" transcendental numbers, though, and many famous constants including "e" and π either are not or are not known to be very well approximable in the above sense [K. Mahler,"On the Approximation of π", Nederl. Akad. Wetensch. Proc. Ser. A. 56; Indagationes Math. 15, (1953), pp.30–42.] .

Auxiliary functions: Hermite to Baker

Fortunately other methods were pioneered in the nineteenth century to deal with the algebraic properties of "e", and consequently of π through Euler's identity. This work centred on use of the so-called auxiliary function. These are functions which typically have many zeros at the points under consideration. Here "many zeros" may mean literally a lot of zeros, or as few as one zero but with a high multiplicity, or even many zeros all with high multiplicity. Charles Hermite used auxiliary functions that approximated the functions "e"^"kx" for each natural number "k" in order to prove the transcendence of "e" in 1873 [C. Hermite, "Sur la fonction exponentielle", C. R. Acad. Sci. Paris 77, (1873).] . His work was built upon by Ferdinand von Lindemann in the 1880s [F. Lindemann, "Über die Zahl π", Mathematische Annalen 20, (1882), pp.213–225.] in order to prove that "e"^α is transcendental for nonzero algebraic numbers α. In particular this proved that π is transcendental since "e"^π"i" is algebraic, and thus answered the problem of antiquity as to whether it was possible to square the circle. Karl Weierstrass developed their work yet further and eventually proved the Lindemann-Weierstrass theorem in 1885 [K. Weierstrass, "Zu Hrn. Lindemann's Abhandlung: 'Über die Ludolph'sche Zahl' ", Sitzungber. Königl. Preuss. Akad. Wissensch. zu Berlin 2, (1885), pp.1067–1086.] .

In 1900 David Hilbert posed his famous collection of problems. The seventh of these, and one of the hardest in Hilbert's estimation, asked about the transcendence of numbers of the form "a"^"b" where "a" and "b" are algebraic, "a" isn't zero or one, and "b" is irrational. In the 1930s Alexander Gelfond [A.O. Gel'fond, "Sur le septième Problème de D. Hilbert", Izv. Akad. Nauk SSSR 7, (1934), pp.623–630.] and Theodor Schneider [T. Schneider, "Transzendenzuntersuchungen periodischer Funktionen. I. Transzendend von Potenzen", J. reine angew. Math. 172, (1934), pp.65–69.] proved that all such numbers were indeed transcendental using a non-explicit auxiliary function whose existence was granted by Siegel's lemma. This result, the Gelfond-Schneider theorem, proved the transcendence of numbers such as "e"^π and the Gelfond–Schneider constant.

The next big result in this field had to wait until the 1960s, when Alan Baker made progress on a problem posed by Gel'fond on linear forms in logarithms. Gel'fond himself had managed to find a non-trivial lower bound for the quantity: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 Gel'fond, though. Baker succeeded in finding such bounds, solving Gauss' class number problem for class number one in the process. This work won Baker the Fields medal for its uses in solving Diophantine 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" are linearly independent over the rational numbers, then the number:is transcendental. [A. Baker, "Linear forms in the logarithms of algebraic numbers. I, II, III", Mathematika 13 ,(1966), pp.204–216; ibid. 14, (1967), pp.102–107; ibid. 14, (1967), pp.220–228, MathSciNet | id = 0220680]

Other techniques: Cantor and Zilber

Liouville proved that in the 1850s that transcendental numbers exist and even gave examples, but his ideas can be used to give only a countable number of transcendental numbers. Indeed, when he published his results the notion of countability and uncountability had not yet been developed. In the 1870s Georg Cantor started to develop set theory and in 1874 published a paper proving that the algebraic numbers could be put in one-to-one correspondence with the set of natural numbers, and thus that the set of transcendental numbers must be uncountable. [G. Cantor, "Über eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen", J. Reine Angew. Math. 77, (1874), pp.258–262.] Later, in 1891, Cantor used his more familiar diagonal argument to prove the same result. [G. Cantor, "Über eine elementare Frage der Mannigfaltigkeitslehre", Jahresbericht der Detschen Mathematiker-Vereinigung 1, (1891), pp.75–78.] While Cantor's result is often quoted as being purely existential and thus unusable for constructing a single transcendental number [M. Kac and U. Stanislaw, "Mathematics and Logic", Fredering A. Praeger, (1968), p.13.] [E.T. Bell, "Men of Mathematics", New York : Simon & Schuster, (1937), p.569.] , the proofs in both the aforementioned papers give methods to construct transcendental numbers. [R. Gray, "Georg Cantor and transcendental numbers", Amer. Math. Monthly 101, (1994), no.9, pp.819–832, MathSciNet | id = 1300488]

While Cantor used set theory to prove the plenitude of transcendental numbers, a recent development has been the use of model theory in attempts to prove an unsolved problem in transcendental number theory. The problem is to determine the transcendence degree of the field:$K=mathbb\{Q\}(x\_1,ldots,x\_n,e^\{x\_1\},ldots,e^\{x\_n\})$for complex numbers "x"₁,...,"x"_"n" that are linearly independent over the rational numbers. Stephen Schanuel conjectured that the answer is "n", 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 the complex numbers equipped with the operations of addition, multiplication, and exponentiation. Moreover, in this abstract structure Schanuel's conjecture does indeed hold. [B. Zilber, "Pseudo-exponentiation on algebraically closed fields of characteristic zero", Annals of Pure and Applied Logic, 132, (2005), pp.67–95, MathSciNet | id = 2102856] Unfortunately it is not yet known that this structure is in fact the same as the complex numbers with the operations mentioned, it could be that Schanuel's conjecture is false and that there exists some other abstract structure that behaves very similarly to the complex numbers but where Schanuel's conjecture holds. Zilber did provide several criteria that would prove the structure in question was C, but could not prove the so-called Strong Exponential Closure axiom. The simplest case of this axiom has since been proved [D. Marker, "A remark on Zilber’s pseudoexponentiation", J. Symbolic Logic Volume 71, Issue 3 ,(2006), pp.791–798, MathSciNet | id = 2250821] , but a proof that it holds in full generality is required to complete the proof of the conjecture.

Approaches

A typical problem in this area of mathematics is to work out whether a given number is transcendental. However this problem is in general extremely difficult. Cantor used a cardinality argument to show that there are only countably many algebraic numbers, and hence almost all numbers are transcendental. But despite this it is extremely difficult to actually prove that a given number is transcendental.

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 if "P"(α)≠0 for every non-zero polynomial "P" with integer coefficents, this problem can be approached by trying to find lower bounds of the form

:|"P"(α)| > "F"("A","d")

where the right hand side is some positive function depending on the some measure "A" of the size of the coefficients of "P", and its degree "d", and such that these lower bounds apply to all "P" ≠ 0. Such a bound is called a transcendence measure.

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

:|"ax" + "b"|.

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

Major Results

The Gelfond-Schneider theorem was the major advance in transcendence theory in the period 1900-1950. In the 1960s the method of Alan Baker on linear forms in logarithms of algebraic numbers reanimated transcendence theory, with applications to numerous classical problems and diophantine equations.

Open Problems

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 found here.

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 that "e" and "π" are transcendental that doesn't imply that "e"+"π" is transcendental, nor other combinations of the two. Another major problem is dealing with numbers that are not related to the exponential function. The main results in transcendence theory tend to revolve around "e" 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 that "e"+"π" is transcendental. It still revolves around the exponential function however and so would not necessarily deal with numbers such as Apéry's constant or the Euler–Mascheroni constant. Another extremely difficult unsolved problem is the so-called Constant or Identity problem [D. Richardson, "Some Undecidable Problems Involving Elementary Functions of a Real Variable", J. Symbolic Logic 33, (1968), pp.514–520, MathSciNet | id = 0239976.] .

Notes

References

* A.O. Gelfond, "Transcendental and Algebraic Numbers," Dover Publications Inc., 1960.
* Serge Lang, "Introduction to Transcendental Numbers," Addison-Wesley Publishing Company, 1966.
* Alan Baker, "Transcendental Number Theory," Cambridge University Press, 1975.