Schanuel's conjecture#Other consequences

Schanuel's conjecture can be given as follows:{{Cite web |last=Waldschmidt |first=Michel |date=2021 |title=Schanuel's Conjecture: algebraic independence of transcendental numbers |url=https://webusers.imj-prg.fr/~michel.waldschmidt/articles/pdf/SchanuelEn.pdf}}

{{Math theorem

| math_statement = Given any set of $n$ complex numbers $\backslash \{z\_1,...,z\_n\backslash \}$ that are linearly independent over $\backslash mathbb\; Q$, the field extension $\backslash mathbb\; Q\; (z\_1,...,z\_n,e^\{z\_1\},...,e^\{z\_n\})$ has transcendence degree at least $n$ over $\backslash mathbb\; Q$.

| name = Schanuel's conjecture

}}

Schanuel's conjecture, if proven, would generalize most known results in transcendental number theory and establish a large class of numbers transcendental. Special cases of Schanuel's conjecture include:

{{Main|Lindemann–Weierstrass theorem}}

Considering Schanuel's conjecture for only $n=1$ gives that for nonzero complex numbers $z$, at least one of the numbers $z$ and $e^z$ must be transcendental. This was proved by Ferdinand von Lindemann in 1882.{{Cite journal |last=von Lindemann |first=Ferdinand |date=1882 |title=Ueber die Zahl π |url=http://gdz.sub.uni-goettingen.de/dms/resolveppn/?PPN=GDZPPN002246910 |journal=Mathematische Annalen |volume=20 |pages=213–225 |issn=0025-5831}}

If the numbers $z\_1,...,z\_n$ are taken to be all algebraic and linearly independent over $\backslash mathbb\; Q$ then the $e^\{z\_1\},...,e^\{z\_n\}$result to be transcendental and algebraically independent over $\backslash mathbb\; Q$. The first proof for this more general result was given by Carl Weierstrass in 1885.{{Harvnb|Weierstrass|1885|pp=1067–1086}},

This so-called Lindemann–Weierstrass theorem implies the transcendence of the numbers E (mathematical constant) and {{pi}}. It also follows that for algebraic numbers $\backslash alpha$ not equal to 0 or 1, both $e^\backslash alpha$ and $\backslash ln(\backslash alpha)$ are transcendental. It further gives the transcendence of the trigonometric functions at nonzero algebraic values.

{{Main|Baker's theorem}}

Another special case was proved by Alan Baker in 1966: If complex numbers $\backslash lambda\_1,...,\backslash lambda\_n$ are chosen to be linearly independent over the rational numbers $\backslash mathbb\; Q$ such that $e^\{\backslash lambda\_1\},...,e^\{\backslash lambda\_n\}$ are algebraic, then $\backslash lambda\_1,...,\backslash lambda\_n$ are also linearly independent over the algebraic numbers $\backslash mathbb\; \backslash overline\{Q\}$.

Schanuel's conjecture would strengthen this result, implying that $\backslash lambda\_1,...,\backslash lambda\_n$ would also be algebraically independent over $\backslash mathbb\; Q$ (and equivalently over $\backslash mathbb\; \backslash overline\{Q\}$).

{{Main|Gelfond–Schneider theorem}}

In 1934 it was proved by Aleksander Gelfond and Theodor Schneider that if $\backslash alpha$ and $\backslash beta$ are two algebraic complex numbers with $\backslash alpha\backslash notin\backslash \{0,1\backslash \}$ and $\backslash beta\backslash notin\; \backslash mathbb\; Q$, then $\backslash alpha^\backslash beta$ is transcendental.

This establishes the transcendence of numbers like Hilbert's constant $2^\backslash sqrt\{2\}$ and Gelfond's constant $e^\backslash pi$.{{Cite web |last=Weisstein |first=Eric W. |title=Gelfond's Theorem |url=https://mathworld.wolfram.com/GelfondsTheorem.html |access-date=2024-10-26 |website=mathworld.wolfram.com |language=en}}

The Gelfond–Schneider theorem follows from Schanuel's conjecture by setting $n=3$ and $z\_1=\backslash beta,\; z\_2=\backslash ln\backslash alpha,\; z\_3=\backslash beta\; \backslash ln\backslash alpha$. It also would follow from the strengthened version of Baker's theorem above.

{{Main|Four exponentials conjecture}}

The currently unproven four exponentials conjecture would also follow from Schanuel's conjecture: If $z\_1,z\_2$ and $w\_1,w\_2$ are two pairs of complex numbers, with each pair being linearly independent over the rational numbers, then at least one of the following four numbers is transcendental:

$$e^\{z\_1w\_1\},\; e^\{z\_1w\_2\},\; e^\{z\_2w\_1\},\; e^\{z\_2w\_2\}.$$

The four exponential conjecture would imply that for any irrational number $t$, at least one of the numbers $2^t$ and $3^t$ is transcendental. It also implies that if $t$ is a positive real number such that both $2^t$ and $3^t$ are integers, then $t$ itself must be an integer. The related six exponentials theorem has been proven.

Schanuel's conjecture, if proved, would also establish many nontrivial combinations of {{Math|{{var|e}}}}, {{pi}}, algebraic numbers and elementary functions to be transcendental:{{Cite book |last1=Murty |first1=M. Ram |url=https://link.springer.com/book/10.1007/978-1-4939-0832-5 |title=Transcendental Numbers |last2=Rath |first2=Purusottam |date=2014 |publisher=Springer |isbn=978-1-4939-0832-5 |language=en |doi=10.1007/978-1-4939-0832-5}}{{cite arXiv |last1=Cheng |first1=Chuangxun |title=Some consequences of Schanuel's Conjecture |date=2008-05-07 |last2=Dietel |first2=Brian |last3=Herblot |first3=Mathilde |last4=Huang |first4=Jingjing |last5=Krieger |first5=Holly |last6=Marques |first6=Diego |last7=Mason |first7=Jonathan |last8=Mereb |first8=Martin |last9=Wilson |first9=S. Robert|class=math.NT |eprint=0804.3550 }}

$$e+\backslash pi,\; e\backslash pi,\; e^\{\backslash pi^2\},\; e^e,\; \backslash pi^e,\; \backslash pi^\backslash sqrt\{2\},\; \backslash pi^\backslash pi,\; \backslash pi^\{\backslash pi^\backslash pi\},\; \backslash ,\; \backslash log\backslash pi,\; \backslash ,\; \backslash log\backslash log2,\; \backslash ,\; \backslash sin\; e,\; ...$$

In particular it would follow that {{Math|{{var|e}}}} and {{pi}} are algebraically independent simply by setting $z\_1=1$ and $z\_2=i\backslash pi$.

Euler's identity states that $e^\{i\backslash pi\}+1=0$. If Schanuel's conjecture is true then this is, in some precise sense involving exponential rings, the only non-trivial relation between {{Math|{{var|e}}}}, {{pi}}, and {{Math|{{var|i}}}} over the complex numbers.{{Cite journal |first=Giuseppina |last=Terzo |title=Some consequences of Schanuel's conjecture in exponential rings |journal=Communications in Algebra |volume=36 |issue=3 |pages=1171–1189 |doi=10.1080/00927870701410694 |year=2008 |s2cid=122764821 }}

The converse Schanuel conjectureScott W. Williams, [http://www.math.buffalo.edu/~sww/0papers/million.buck.problems.mi.pdf Million Bucks Problems] is the following statement:

{{block indent|1=Suppose F is a countable field with characteristic 0, and e : F → F is a homomorphism from the additive group (F,+) to the multiplicative group (F,·) whose kernel is cyclic. Suppose further that for any n elements x₁,...,x_n of F which are linearly independent over $\backslash mathbb\{Q\}$, the extension field $\backslash mathbb\{Q\}$(x₁,...,x_n,e(x₁),...,e(x_n)) has transcendence degree at least n over $\backslash mathbb\{Q\}$. Then there exists a field homomorphism h : F → $\backslash mathbb\{C\}$ such that h(e(x)) = exp(h(x)) for all x in F.}}

A version of Schanuel's conjecture for formal power series, also by Schanuel, was proven by James Ax in 1971.{{Cite journal |first=James |last=Ax |title=On Schanuel's conjectures |journal=Annals of Mathematics |issue=2 |volume=93 |year=1971 |pages=252–268 |jstor=1970774|doi=10.2307/1970774 }} It states:

{{block indent|1=Given any n formal power series f₁,...,f_n in t$\backslash mathbb\{C\}$t which are linearly independent over $\backslash mathbb\{Q\}$, then the field extension $\backslash mathbb\{C\}$(t,f₁,...,f_n,exp(f₁),...,exp(f_n)) has transcendence degree at least n over $\backslash mathbb\{C\}$(t).}}

Although ostensibly a problem in number theory, Schanuel's 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, $\backslash mathbb\{R\}$_exp, is decidable provided Schanuel's conjecture is true.{{Cite book |last1=Macintyre |first1=A. |title=Kreiseliana: About and Around Georg Kreisel |last2=Wilkie |first2=A. J. |publisher=Peters |year=1996 |isbn=978-1-56881-061-4 |editor-last=Odifreddi |editor-first=Piergiorgio |location=Wellesley |pages=441–467 |chapter=On the decidability of the real exponential field |name-list-style=amp}} In fact, to prove this result, they only needed the real version of the conjecture, which is as follows:{{Cite journal |last1=Kirby |first1=Jonathan |last2=Zilber |first2=Boris |name-list-style=amp |year=2006 |title=The uniform Schanuel conjecture over the real numbers |journal=Bull. London Math. Soc. |volume=38 |issue=4 |pages=568–570 |citeseerx=10.1.1.407.5667 |doi=10.1112/S0024609306018510 |s2cid=122077474}}

{{block indent|1=Suppose x₁,...,x_n are real numbers and the transcendence degree of the field $\backslash mathbb\{Q\}$(x₁,...,x_n, exp(x₁),...,exp(x_n)) is strictly less than n, then there are integers m₁,...,m_n, not all zero, such that m₁x₁ +...+ m_nx_n = 0.}}

This would be a positive solution to Tarski's exponential function problem.

A related conjecture called the uniform real Schanuel's conjecture essentially says the same but puts a bound on the integers m_i. 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 $\backslash mathbb\{R\}$_exp. This conjecture states that there is a computable upper bound on the norm of non-singular solutions to systems of exponential polynomials; this is, non-obviously, a consequence of Schanuel's conjecture for the reals.

It is also known that Schanuel's conjecture would be a consequence of conjectural results in the theory of motives. In this setting 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.{{Cite journal |first=Cristiana |last=Bertolin |title=Périodes de 1-motifs et transcendance |journal=Journal of Number Theory |volume=97 |issue=2 |year=2002 |pages=204–221 |doi=10.1016/S0022-314X(02)00002-1 |doi-access=free |hdl=2318/103562 |hdl-access=free }}

While a proof of Schanuel's conjecture seems a long way off,{{Cite book|last=Waldschmidt|first=Michel|title=Diophantine approximation on linear algebraic groups|publisher=Springer|year=2000|isbn=978-3-662-11569-5|location=Berlin}} connections with model theory have prompted a surge of research on the conjecture.

In 2004, Boris Zilber systematically constructed exponential fields K_exp that are algebraically closed and of characteristic zero, and such that one of these fields exists for each uncountable cardinality.{{Cite journal |first=Boris |last=Zilber |title=Pseudo-exponentiation on algebraically closed fields of characteristic zero |journal=Annals of Pure and Applied Logic |volume=132 |year=2004 |issue=1 |pages=67–95 |doi=10.1016/j.apal.2004.07.001 |doi-access=free }} He axiomatised these fields and, using Hrushovski's construction and techniques inspired by work of Shelah on categoricity in infinitary logics, proved that this theory of "pseudo-exponentiation" has a unique model in each uncountable cardinal. Schanuel's conjecture is part of this axiomatisation, and so the natural conjecture that the unique model of cardinality continuum is actually isomorphic to the complex exponential field implies Schanuel's conjecture. In fact, Zilber showed that this conjecture holds if and only if both Schanuel's conjecture and the Exponential-Algebraic Closedness conjecture hold.{{Cite journal |first=Boris |last=Zilber |title=Exponential sums equations and the Schanuel conjecture |journal=J. London Math. Soc. |issue=2 |volume=65 |year=2002 |pages=27–44 |doi= 10.1112/S0024610701002861|s2cid=123143365 }} As this construction can also give models with counterexamples of Schanuel's conjecture, this method cannot prove Schanuel's conjecture.{{cite journal |last1=Bays |first1=Martin |last2=Kirby |first2=Jonathan |title=Pseudo-exponential maps, variants, and quasiminimality |journal=Algebra Number Theory |date=2018 |volume=12 |issue=3 |pages=493–549 |doi=10.2140/ant.2018.12.493 |arxiv=1512.04262 |s2cid=119602079 }}

• Four exponentials conjecture
• Algebraic independence
• List of unsolved problems in mathematics
• Existential Closedness conjecture
• Zilber-Pink conjecture
• Pregeometry

{{Reflist}}

• {{Citation | last=Weierstrass | first=K. | author-link=Karl Weierstrass | year=1885 | title=Zu Lindemann's Abhandlung. "Über die Ludolph'sche Zahl". | journal=Sitzungsberichte der Königlich Preussischen Akademie der Wissen-schaften zu Berlin | volume=5 | pages=1067–1085 | url=https://books.google.com/books?id=jhlEAQAAMAAJ&pg=PA1067}}

• {{MathWorld|urlname=SchanuelsConjecture|title=Schanuel's Conjecture}}

Category:Conjectures

Category:Unsolved problems in number theory

Category:Exponentials

Category:Transcendental numbers