Coase conjecture

In mathematics, specifically transcendental number theory, the six exponentials theorem is a result that, given the right conditions on the exponents, guarantees the transcendence of at least one of a set of exponentials.

Statement

If x₁, x₂,..., x_d are d complex numbers that are linearly independent over the rational numbers, and y₁, y₂,...,y_l are l complex numbers that are also linearly independent over the rational numbers, and if dl > d + l, then at least one of the following dl numbers is transcendental:

$\exp (x_i{y}_j),(1\le i\le d,1\le j\le l).$

The most interesting case is when d = 3 and l = 2, in which case there are six exponentials, hence the name of the result. The theorem is weaker than the related but thus far unproved four exponentials conjecture, whereby the strict inequality dl > d + l is replaced with dl ≥ d + l, thus allowing d = l = 2.

The theorem can be stated in terms of logarithms by introducing the set L of logarithms of algebraic numbers:

$\mathcal{ℒ}=\{\lambda \in \mathbb{ℂ}:e^{\lambda }\in \overline{\mathbb{\mathbb{Q}}}\}.$

The theorem then says that if λ_ij are elements of L for i = 1, 2 and j = 1, 2, 3, such that λ₁₁, λ₁₂, and λ₁₃ are linearly independent over the rational numbers, and λ₁₁ and λ₂₁ are also linearly independent over the rational numbers, then the matrix

$M=\left(\begin{array}{ccc}{\lambda }_{11}& {\lambda }_{12}& {\lambda }_{13}\\ {\lambda }_{21}& {\lambda }_{22}& {\lambda }_{23}\end{array}\right)$

has rank 2.

History

A special case of the result where x₁, x₂, and x₃ are logarithms of positive integers, y₁ = 1, and y₂ is real, was first mentioned in a paper by Leonidas Alaoglu and Paul Erdős from 1944 in which they try to prove that the ratio of consecutive colossally abundant numbers is always prime. They claimed that Carl Ludwig Siegel knew of a proof of this special case, but it is not recorded. Using the special case they manage to prove that the ratio of consecutive colossally abundant numbers is always either a prime or a semiprime.

The theorem was first explicitly stated and proved in its complete form independently by Serge Lang and Kanakanahalli Ramachandra in the 1960s.

Five exponentials theorem

A stronger, related result is the five exponentials theorem, which is as follows. Let x₁, x₂ and y₁, y₂ be two pairs of complex numbers, with each pair being linearly independent over the rational numbers, and let γ be a non-zero algebraic number. Then at least one of the following five numbers is transcendental:

$e^{x_1{y}_1},e^{x_1{y}_2},e^{x_2{y}_1},e^{x_2{y}_2},e^{\gamma x_2/x_1}.$

This theorem implies the six exponentials theorem and in turn is implied by the as yet unproven four exponentials conjecture, which says that in fact one of the first four numbers on this list must be transcendental.

Sharp six exponentials theorem

Another related result that implies both the six exponentials theorem and the five exponentials theorem is the sharp six exponentials theorem. This theorem is as follows. Let x₁, x₂, and x₃ be complex numbers that are linearly independent over the rational numbers, and let y₁ and y₂ be a pair of complex numbers that are linearly independent over the rational numbers, and suppose that β_ij are six algebraic numbers for 1 ≤ i ≤ 3 and 1 ≤ j ≤ 2 such that the following six numbers are algebraic:

$e^{x_1{y}_1-{\beta }_{11}},e^{x_1{y}_2-{\beta }_{12}},e^{x_2{y}_1-{\beta }_{21}},e^{x_2{y}_2-{\beta }_{22}},e^{x_3{y}_1-{\beta }_{31}},e^{x_3{y}_2-{\beta }_{32}}.$

Then x_i y_j = β_ij for 1 ≤ i ≤ 3 and 1 ≤ j ≤ 2. The six exponentials theorem then follows by setting β_ij = 0 for every i and j, while the five exponentials theorem follows by setting x₃ = γ/x₁ and using Baker's theorem to ensure that the x_i are linearly independent.

There is a sharp version of the five exponentials theorem as well, although it as yet unproven so is known as the sharp five exponentials conjecture. This conjecture implies both the sharp six exponentials theorem and the five exponentials theorem, and is stated as follows. Let x₁, x₂ and y₁, y₂ be two pairs of complex numbers, with each pair being linearly independent over the rational numbers, and let α, β₁₁, β₁₂, β₂₁, β₂₂, and γ be six algebraic numbers with γ ≠ 0 such that the following five numbers are algebraic:

$e^{x_1{y}_1-{\beta }_{11}},e^{x_1{y}_2-{\beta }_{12}},e^{x_2{y}_1-{\beta }_{21}},e^{x_2{y}_2-{\beta }_{22}},e^{(\gamma x_2/x_1)-\alpha }.$

Then x_i y_j = β_ij for 1 ≤ i, j ≤ 2 and γx₂ = αx₁.

A consequence of this conjecture that isn't currently known would be the transcendence of e^π², by setting x₁ = y₁ = β₁₁ = 1, x₂ = y₂ = iπ, and all the other values in the statement to be zero.

Strong six exponentials theorem

The logical implications between the various problems in this circle. Those in red are as yet unproven while those in blue are known results. The top most result refers to that discussed at Baker's theorem, while the four exponentials conjectures are detailed at the four exponentials conjecture article.

A further strengthening of the theorems and conjectures in this area are the strong versions. The strong six exponentials theorem is a result proved by Damien Roy that implies the sharp six exponentials theorem. This result concerns the vector space over the algebraic numbers generated by 1 and all logarithms of algebraic numbers, denoted here as L^∗. So L^∗ is the set of all complex numbers of the form

${\beta }_0+\sum _{i=1}^n{\beta }_i\log {\alpha }_i,$

for some n ≥ 0, where all the β_i and α_i are algebraic and every branch of the logarithm is considered. The strong six exponentials theorem then says that if x₁, x₂, and x₃ are complex numbers that are linearly independent over the algebraic numbers, and if y₁ and y₂ are a pair of complex numbers that are also linearly independent over the algebraic numbers then at least one of the six numbers x_i y_j for 1 ≤ i ≤ 3 and 1 ≤ j ≤ 2 is not in L^∗. This is stronger than the standard six exponentials theorem which says that one of these six numbers is not simply the logarithm of an algebraic number.

There is also a strong five exponentials conjecture formulated by Michel Waldschmidt It would imply both, the strong six exponentials theorem and the sharp five exponentials conjecture. This conjecture claims that if x₁, x₂ and y₁, y₂ are two pairs of complex numbers, with each pair being linearly independent over the algebraic numbers, then at least one of the following five numbers is not in L^∗:

$x_1{y}_1,x_1{y}_2,x_2{y}_1,x_2{y}_2,x_1/x_2.$

All the above conjectures and theorems are consequences of the unproven extension of Baker's theorem, that logarithms of algebraic numbers that are linearly independent over the rational numbers are automatically algebraically independent too. The diagram on the right shows the logical implications between all these results.

Generalization to commutative group varieties

The exponential function $e^z$ uniformizes the exponential map of the multiplicative group $G_m$. The six exponential theorem can be therefore reformulated in a more abstract fashion:

Let $\mathbb{ℂ}$ be the field of complex numbers and let $G=G_m^2$. Let $u:\mathbb{ℂ}$→ $G(\mathbb{ℂ})$ be a non-zero complex-analytic group homomorphism. Denote by $L$ the group of numbers $l$ in $\mathbb{ℂ}$, such that $u(l)$ is an algebraic point of $G$. If $L$ can be only generated by more than two elements $x_1,x_2,x_3,...$ over the field of rational numbers $\mathbb{\mathbb{Q}}$, then the image $u(\mathbb{ℂ})$ is an algebraic subgroup of $G(\mathbb{ℂ})$.

In this way, the statement of the six exponentials theorem can be generalized to an arbitrary commutative group variety $G$ over the field of algebraic numbers. Alternatively, one can replace $G=G_m^2$ by $G=G_m^3$ and "more than two elements" by "more than one element" to obtain another variant of the generalization. This generalized six exponential conjecture, however, seems out of scope at the current state of transcendental number theory.

For the special, but interesting cases $G=G_m$× $E$ and $G=E$× $E^{\prime }$, with $E,E^{\prime }$ elliptic curves over the field of algebraic numbers, results towards the generalized six exponential conjecture could be established by Aleksander Momot. These results involve the exponential function $e^z$ and a Weierstrass function $\mathrm{\wp }$ resp. two Weierstrass functions $\mathrm{\wp },{\mathrm{\wp }}^{\prime }$ with algebraic invariants $g_2,g_3,{g_2}^{\prime },{g_3}^{\prime }$, instead of two exponential functions $e^{y_{1}z},e^{y_{2}z}$ as in the classical statement. In the classical statement, the numbers $x_1,x_2,x_3$ play the role of a generating set of $L$, that is, $L=\mathbb{\mathbb{Q}}{x}_1+\mathbb{\mathbb{Q}}{x}_2+\mathbb{\mathbb{Q}}{x}_3$.

For an algebraic group $G=G_m$× $E$ it is proved in, among others, that if $E$ is not isogenous to a curve over a real field and if $u(\mathbb{ℂ})$ is not an algebraic subgroup of $G(\mathbb{ℂ})$, then $L$ can be either generated by two elements $x_1,x_2$ over $\mathbb{\mathbb{Q}}$, or a minimal generating set of $L$ over $\mathbb{\mathbb{Q}}$ consists of three elements $x_1,x_2,x_3$ which are not all contained in a real line $\mathbb{ℝ}c$ ($c$ a non-zero complex number). A similar result is shown for $G=E$ × $E^{\prime }$.

Notes

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References

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  The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
  
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  A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
  
  The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
  
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  I had like 17 domains hosted on single account, and never had any special troubles. If you are not happy with the service you will get your money back with in 45 days, that's guaranteed. But the Search Engine utility inside the Hostgator account furnished an instant score for my launched website. Fantastico is unable to install WordPress in a directory which already have any file i.e to install WordPress using Fantastico the destination directory must be empty and it should not have any previous installation files. When you share great information, others will take note. Once your hosting is purchased, you will need to setup your domain name to point to your hosting. Money Back: All accounts of Hostgator come with a 45 day money back guarantee. If you have any queries relating to where by and how to use Hostgator Discount Coupon, you can make contact with us at our site. If you are starting up a website or don't have too much website traffic coming your way, a shared plan is more than enough. Condition you want to take advantage of the worldwide web you prerequisite a HostGator web page, -1 of the most trusted and unfailing web suppliers on the world wide web today. Since, single server is shared by 700 to 800 websites, you cannot expect much speed.
  
  
  
  Hostgator tutorials on how to install Wordpress need not be complicated, especially when you will be dealing with a web hosting service that is friendly for novice webmasters and a blogging platform that is as intuitive as riding a bike. After that you can get Hostgator to host your domain and use the wordpress to do the blogging. Once you start site flipping, trust me you will not be able to stop. I cut my webmaster teeth on Control Panel many years ago, but since had left for other hosting companies with more commercial (cough, cough) interfaces. If you don't like it, you can chalk it up to experience and go on. First, find a good starter template design. When I signed up, I did a search for current "HostGator codes" on the web, which enabled me to receive a one-word entry for a discount. Your posts, comments, and pictures will all be imported into your new WordPress blog.
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