13 Consequences in transcendence degree one
Results of the library’s work on Diaz’s conjecture that follow from the four exponentials theorem in transcendence degree one (Theorem 6.31): a \(2\times 2\) matrix with non-zero entries in \(\mathcal L\), the logarithms of algebraic numbers, that generate a field of transcendence degree at most one and with vanishing determinant has rationally dependent rows or columns. Each statement here is one matrix away from that theorem, or from Brownawell’s corollaries of 1974 [ Bro74 ] ; they were not found in the sources read in this form, and they are short (Chapter 14). The last section holds a conditional statement: Waldschmidt’s strong five exponentials conjecture implies Diaz’s.
13.1 Products and quotients on an axis
Let \(\lambda ,\mu \in \mathcal L\setminus \{ 0\} \) with \(\operatorname {trdeg}_{\mathbb {Q}}\mathbb {Q}(\lambda ,\mu ,\bar\lambda ,\bar\mu )\le 1\). If \(\lambda \mu \) is real, then \(\lambda \) and \(\mu \) are both real, or both purely imaginary, or \(\mu \in \mathbb {Q}\bar\lambda \).
Source: A step of Theorem 13.3.
The matrix \(\begin{pmatrix} \lambda & \bar\lambda \\ \bar\mu & \mu \end{pmatrix}\) has non-zero entries in \(\mathcal L\) and determinant \(\lambda \mu -\overline{\lambda \mu }=0\), so Theorem 6.31 applies. A row relation gives \(\mu \in \mathbb {Q}\bar\lambda \). A column relation gives \(\bar\lambda =q\lambda \) and \(\mu =q\bar\mu \) with \(q\in \mathbb {Q}\), and \(|q|=1\) forces \(q=\pm 1\).
Let \(\lambda ,\mu \in \mathcal L\setminus \{ 0\} \) with \(\operatorname {trdeg}_{\mathbb {Q}}\mathbb {Q}(\lambda ,\mu ,\bar\lambda ,\bar\mu )\le 1\). If \(\lambda \mu \) is purely imaginary, then one of \(\lambda ,\mu \) is real and the other purely imaginary.
Source: A step of Theorem 13.3.
The matrix \(\begin{pmatrix} \lambda & \bar\lambda \\ -\bar\mu & \mu \end{pmatrix}\) has determinant \(\lambda \mu +\overline{\lambda \mu }=0\). By Theorem 6.31, a row relation would force \(\mu =0\), and a column relation puts \(\lambda \) and \(\mu \) on different axes.
Let \(\ell _0,\ell _1\in \mathcal L\), neither real nor purely imaginary, with \(\operatorname {trdeg}_{\mathbb {Q}}\mathbb {Q}(\ell _0,\ell _1,\bar\ell _0,\bar\ell _1)\le 1\). If \(\ell _1/\ell _0\) is real or purely imaginary, then \(\ell _1/\ell _0\in \mathbb {Q}\).
Source: Not found in the sources read, but short (Chapter 14). The statement is Diaz’s conjecture (Qr2) [ Dia07 , p. 376 ] , here in transcendence degree one; Diaz derives (Qr2) from the four exponentials conjecture with the same matrix [ Dia07 , p. 377 ] . It is one substitution in [ Bro74 , Corollary 7 ] , at \((\ell _1/\ell _0,\bar\ell _0/\ell _0,\ell _0)\).
Take \(\lambda =\bar\ell _0\) and \(\mu =\ell _1\), so that \(\lambda \mu =|\ell _0|^{2}\, \ell _1/\ell _0\) lies on an axis exactly when \(\ell _1/\ell _0\) does. Proposition 13.2 rules out the imaginary axis, since neither \(\ell _0\) nor \(\ell _1\) lies on an axis; on the real axis, Proposition 13.1 leaves only \(\ell _1\in \mathbb {Q}\ell _0\).
13.2 Moduli in a rational ratio
Let \(u,v\in \mathcal L\setminus \{ 0\} \) with \(|u|^{2}/|v|^{2}\in \mathbb {Q}\) and \(\operatorname {trdeg}_{\mathbb {Q}}\mathbb {Q}(u,v,\bar u,\bar v)\le 1\). Then \(v\in \mathbb {Q}u\) or \(v\in \mathbb {Q}\bar u\).
Source: Not found in the sources read, but short (Chapter 14). It is [ Bro74 , Corollary 7 ] at \((v/u,c\bar v/u,u)\), word for word.
With \(c=|u|^{2}/|v|^{2}\), the matrix \(\begin{pmatrix} u & v \\ c\bar v & \bar u \end{pmatrix}\) has non-zero entries in \(\mathcal L\) and determinant \(|u|^{2}-c|v|^{2}=0\). By Theorem 6.31, a column relation gives \(v\in \mathbb {Q}u\) and a row relation \(v\in \mathbb {Q}\bar u\).
The conjugates \(\bar u=|u|^{2}/u\) and \(\bar v=|v|^{2}/v\) are algebraic over \(\mathbb {Q}(\pi )\) as well, so \(\mathbb {Q}(u,v,\bar u,\bar v)\) has transcendence degree at most one, and \(|u|^{2}/|v|^{2}\) is rational: Theorem 13.4 applies.
The numbers \(u=t_0+i\pi \) and \(v=t_1+i\pi \) lie in \(\mathcal L\setminus \{ 0\} \), and they are algebraic over \(\mathbb {Q}(\pi )\) because \(t_k^{2}=(t_k^{2}+\pi ^{2})-\pi ^{2}\). Corollary 13.5 gives \(v=qu\) or \(v=q\bar u\) with \(q\in \mathbb {Q}\), and the imaginary parts force \(q=\pm 1\).
The numbers \((\log 2)^{2}+\pi ^{2}\) and \((\log 3)^{2}+\pi ^{2}\) are not both rational.
Source: Not found in the sources read, but short (Chapter 14).
Corollary 13.6 with \(t_0=\log 2\) and \(t_1=\log 3\), since \(\log 3\neq \pm \log 2\).
13.3 Rational data on the real axis
Let \(w\neq 0\) and \(z\notin \mathbb {Q}\) with \(\operatorname {trdeg}_{\mathbb {Q}}\mathbb {Q}(w,z)\le 1\). Then \(w\), \(wz\) and \(wz^{2}\) are not all in \(\mathcal L\).
Source: A step of Theorem 13.9.
Otherwise the matrix \(\begin{pmatrix} w & wz \\ wz & wz^{2} \end{pmatrix}\) has vanishing determinant and entries in \(\mathcal L\) generating a field of transcendence degree at most one, so by Theorem 6.31 its rows or columns satisfy a relation \(pw+q\, wz=0\) with \(p,q\in \mathbb {Q}\) not both zero; this forces \(z=-p/q\in \mathbb {Q}\).
Let \(t\neq 0\) be real with \(e^{t}\) algebraic, and suppose that \(\rho =t^{2}+\pi ^{2}\) is algebraic. Then \(e^{it^{2}/\pi }\), \(e^{\pi ^{2}/t}\) and \(e^{\rho /(i\pi )}\) are transcendental.
Source: The first number is [ Bro74 , Corollary 5 ] at \(\eta =t/\pi \).
The hypothesis makes \(t\) algebraic over \(\mathbb {Q}(\pi )\), since \(t^{2}=\rho -\pi ^{2}\). Proposition 13.8 applies to \((w,z)=(i\pi ,t/(i\pi ))\), where \(e^{w}=-1\) and \(wz=t\), so \(wz^{2}=-it^{2}/\pi \notin \mathcal L\); and to \((w,z)=(t,i\pi /t)\), where \(wz=i\pi \), so \(wz^{2}=-\pi ^{2}/t\notin \mathcal L\). Finally \(\rho /(i\pi )=-it^{2}/\pi -i\pi \), so \(e^{\rho /(i\pi )}=-e^{-it^{2}/\pi }\).
Let \(t\neq 0\) be real with \(e^{t}\) algebraic and \(t^{2}+\pi ^{2}\in \mathbb {Q}\). Then \(e^{i\gamma /\pi }\) is transcendental for every \(\gamma \in \mathbb {Q}\), \(\gamma \neq 0\).
Source: Not found in the sources read, but short (Chapter 14): the two open boundary statements of the library cannot both fail at rational data. The key step is [ Bro74 , Corollary 5 ] at \(\eta =t/\pi \).
Put \(r=t^{2}+\pi ^{2}\). By Theorem 13.9, \(e^{r/(i\pi )}=e^{-ir/\pi }\) is transcendental. If \(e^{i\gamma /\pi }\) were algebraic, write \(r/\gamma =a/b\) with \(a\in \mathbb {Z}\) and \(b\ge 1\); then \((e^{ir/\pi })^{b}=(e^{i\gamma /\pi })^{a}\) would be algebraic, and with it \(e^{ir/\pi }\) and its inverse.
13.4 The strong five exponentials conjecture
Assume Waldschmidt’s strong five exponentials conjecture: if \(x_1,x_2\) are linearly independent over \(\mathbb {Q}\), \(y_1,y_2\) are linearly independent over \(\mathbb {Q}\), \(\eta \neq 0\), the \(\alpha _{ij}\) and \(\beta \) are algebraic, and the five numbers \(e^{x_iy_j-\alpha _{ij}}\) and \(e^{\eta x_2/x_1-\beta }\) are algebraic, then \(x_iy_j=\alpha _{ij}\) for all \(i,j\) and \(\eta x_2=\beta x_1\). Then for every \(u\neq 0\) with \(|u|\) algebraic, \(e^{u}\) is transcendental.
Source: Not found in the sources read, but short (Chapter 14): the derivation is one line. The conjecture is [ Wal04 , Conjecture 3.5 ] .
Suppose \(e^{u}\) algebraic. Then \(u\) is transcendental by Theorem 3.2, so \((1,u)\) and \((1,\bar u)\) are free over \(\mathbb {Q}\), and \(e^{\bar u}=\overline{e^{u}}\) is algebraic. Apply the conjecture to \(x=(1,u)\), \(y=(1,\bar u)\), with \(\alpha =(1,0,0,u\bar u)\), \(\eta =1\) and \(\beta =0\): the five exponentials are \(1\), \(e^{\bar u}\), \(e^{u}\), \(1\) and \(e^{u}\), all algebraic. The conclusion \(x_1y_2=\alpha _{12}\) says \(\bar u=0\), a contradiction.
13.5 Two candidates
Let \(u,v\in \mathbb {C}\) with \(u\bar u\) algebraic. If \(v=cu\) or \(v=c\bar u\) for some \(c\in \mathbb {Q}^{\times }\), then \(u\) and \(v\) are algebraically dependent over \(\mathbb {Q}\).
Source: Elementary.
\(v-cu=0\) is a relation; if \(v=c\bar u\), then \(uv=c\, u\bar u\) is algebraic, which is a relation too.
Let \(u\neq 0\). Then \(|u|\) is algebraic if and only if \(u\bar u\) is, and then \(u\bar u=\rho \) for a positive real algebraic \(\rho \).
Source: Elementary.
\(u\bar u=|u|^{2}\), and \(|u|=\sqrt{u\bar u}\) is algebraic when \(u\bar u\) is.
If \(\bar u=\pm u\) and \(|u|\) is algebraic, then \(u\) is algebraic.
Source: Known: the first step of Case 1 of the proof of [ Dia04 , Proposition 1 ] , p. 551.
\(u=\pm |u|\) or \(u=\pm i|u|\).
Let \(u\neq 0\) be real or purely imaginary with \(|u|\) algebraic. Then \(e^{u}\) is transcendental.
Source: A consequence of Hermite–Lindemann (Theorem 3.2).
On either axis \(u\in \{ \pm |u|,\pm i|u|\} \) is a non-zero algebraic number.
Let \(u,v\) be candidates with \(|v|^{2}/|u|^{2}\in \mathbb {Q}\). Then \(v\in \mathbb {Q}^{\times }u\cup \mathbb {Q}^{\times }\bar u\) if and only if \(u\) and \(v\) are algebraically dependent over \(\mathbb {Q}\).
Source: From the author’s unpublished manuscript on the conjecture, where it was derived from [ RW97 , Théorème 0.2 ] ; it is a direct instance of [ Wal73 , Corollaire 4 ] .
Let \(u\neq v\) be candidates with \(v-u\) real or purely imaginary. Then \(|v|^{2}/|u|^{2}\in \mathbb {Q}\) if and only if \(|v|=|u|\), and \(|v|=|u|\) if and only if \(v=\bar u\) when \(v-u\in i\mathbb {R}\) and \(v=-\bar u\) when \(v-u\in \mathbb {R}\).
Source: From the author’s unpublished manuscript on the conjecture, where it was derived from [ RW97 , Théorème 0.2 ] ; it is a direct instance of [ Wal73 , Corollaire 4 ] .
\(v-\bar v=u-\bar u\) or \(v+\bar v=u+\bar u\) makes \(v\) a root of a quadratic over \(\mathbb {Q}(u)\), so \(u,v\) are dependent, and Corollary 13.16 gives \(v=cu\) or \(v=c\bar u\), \(c\in \mathbb {Q}^{\times }\). A candidate is off both axes (Lemma 13.15), so \(c=1\) in the first case, which is excluded, and \(c=\pm 1\) in the second.
Let \(u\) be a candidate and \(\mu \in \mathcal L\) with \(\mu \notin \mathbb {Q}u\cup \mathbb {Q}\bar u\). Then \(\operatorname {trdeg}_{\mathbb {Q}}\mathbb {Q}(u,\mu ,e^{u\bar u/\mu })\ge 2\).
Source: From the author’s unpublished manuscript on the conjecture; it is [ Wal73 , Corollaire 4 ] at \((u,\bar u,\mu )\).
Theorem 7.11 at \(x=(u,\mu )\), \(y=(\bar u/\mu ,1)\): the column \(y_2=1\) carries \(e^{u}\) and \(e^{\mu }\), both algebraic, so two of the eight numbers are algebraically independent, and all eight are algebraic over \(\mathbb {Q}(u,\mu ,e^{u\bar u/\mu })\).
13.6 Kirby’s weak Schanuel conjecture
The following are equivalent: (i) for every \(u\neq 0\) off both axes with \(|u|\) algebraic, \(e^{u}\) real and \(e^{u}\neq 1\), \(e^{u}\) is transcendental; (ii) for every real \(t\neq 0\) with \(e^{t}\) algebraic, \(t^{2}+\pi ^{2}\) is transcendental.
Source: Elementary, from the quantisation of the imaginary part.
If \(e^{u}\) is real and algebraic, \(u=t+ik\pi \) with \(k\neq 0\), and \(u/k\) has \(\operatorname {Im}=\pi \); conversely \(u=t+i\pi \).
Assume the case \(n=2\) of Kirby’s weak Schanuel conjecture [ Kir18 , Conjecture 1.5 ] : if \(\operatorname {trdeg}_{\mathbb {Q}}\mathbb {Q}(a,b,e^{a},e^{b})\lt 2\), then \(ma+nb\in 2\pi i\mathbb {Z}\) for some integers \((m,n)\neq (0,0)\). Then every candidate \(u\) has \(\operatorname {Re}u\neq 0\) and \(\operatorname {Im}u\in \pi \mathbb {Q}^{\times }\).
Source: Not found in the sources read; short (Chapter 14). It is not the “weak Schanuel” of Calegari and Mazur, which is the algebraic independence of logarithms.
\(\operatorname {trdeg}\mathbb {Q}(u,\bar u,e^{u},e^{\bar u})\le 1\), so \(mu+n\bar u=2\pi ik\); off the axes (Lemma 13.15) this forces \(n=-m\neq 0\) and \(\operatorname {Im}u=(k/m)\pi \).
Assume the case \(n=2\) of Kirby’s weak Schanuel conjecture [ Kir18 , Conjecture 1.5 ] : if \(\operatorname {trdeg}_{\mathbb {Q}}\mathbb {Q}(a,b,e^{a},e^{b})\lt 2\), then \(ma+nb\in 2\pi i\mathbb {Z}\) for some integers \((m,n)\neq (0,0)\). Then Diaz’s conjecture holds if and only if \(t^{2}+\pi ^{2}\) is transcendental for every real \(t\neq 0\) with \(e^{t}\) algebraic.
Source: Not found in the sources read; short (Chapter 14).