10 Rank-one configurations near a point of a circle
Results of the library’s work on Diaz’s conjecture, about the hypothesis that rank-one statements need. The six exponentials theorem (Chapter 5) and Roy’s strong six exponentials theorem [ Roy92 ] take as input a configuration: \(x_1,x_2\) linearly independent over \(\overline{\mathbb {Q}}\) and \(y_1,y_2,y_3\) linearly independent over \(\overline{\mathbb {Q}}\), with all six products \(x_iy_j\) in a given \(\overline{\mathbb {Q}}\)-vector space. Near a point \(u\) of a circle of algebraic radius, that is \(u\neq 0\) with \(u\bar u\) algebraic, Theorem 10.1 shows that generic data carry no configuration, whatever numbers are added, and Theorem 10.6 classifies the four-dimensional extensions \(\overline{\mathbb {Q}}+\overline{\mathbb {Q}}u+\overline{\mathbb {Q}}\bar u+\overline{\mathbb {Q}}z\) that carry one. Neither statement involves \(e^{u}\), so both keep their content if Diaz’s conjecture holds. Neither was found in the sources read; the printed special cases are named in each source line. A third section adds two families of five-dimensional spaces \(\overline{\mathbb {Q}}+\overline{\mathbb {Q}}u+\overline{\mathbb {Q}}\bar u+\overline{\mathbb {Q}}z+\overline{\mathbb {Q}}\bar z\) that carry a configuration invisible to those extensions (Theorem 10.9); that was not found either, but it is short, and its consequences at a candidate are printed (Corollary 10.13). The last section shows that in the power hulls \(\overline{\mathbb {Q}}u^{-k}+\overline{\mathbb {Q}}u^{-1}+\overline{\mathbb {Q}}+\overline{\mathbb {Q}}u+\overline{\mathbb {Q}}u^{k}\) a configuration exists only for \(k=2\) and \(k=3\) (Theorem 10.14), so that the strong six exponentials route at a candidate stops at \(u^{3}\). The last two sections put these results into one mechanism over any field: separation (Theorem 10.16), the normal form of \(2\times 2\) configurations (Theorem 10.22) and configurations in Laurent hulls (Theorem 10.28).
10.1 Generic data
Let \(u\neq 0\) with \(u\bar u\) algebraic, and let \(w_1,\dots ,w_m\in \mathbb {C}\) be such that \(u,w_1,\dots ,w_m\) are algebraically independent over \(\overline{\mathbb {Q}}\). Then there are no \(x_1,x_2\), linearly independent over \(\overline{\mathbb {Q}}\), and \(y_1,y_2,y_3\), linearly independent over \(\overline{\mathbb {Q}}\), with all six products \(x_iy_j\) in \(\overline{\mathbb {Q}}+\overline{\mathbb {Q}}u+\overline{\mathbb {Q}}\bar u+\sum _j\overline{\mathbb {Q}}w_j\).
Source: Not found in the sources read, and not routine (Chapter 14). The case \(m=0\) is [ Roy95 , Theorem 3.4 ] and [ Wal00 , Lemma 12.16 and Exercise 12.10 ] ; see also [ Roy95 , § 3.2 ] and [ Fis01 , p. 186 ] . The general statement was not found in the sources read.
Multiply by \(u\). Since \(\bar u=\rho /u\) with \(\rho \) algebraic, every element of the space times \(u\) is the value at \((u,w)\) of a polynomial over \(\overline{\mathbb {Q}}\) of total degree at most two that is constant once its first variable \(X_0\) is set to \(0\), and evaluation at \((u,w)\) is injective. A configuration therefore gives a rank-one \(2\times 3\) matrix of such polynomials, which factors as \((h,g)\otimes (c_0,c_1,c_2)\) with both factors free. Total degrees add, and \(X_0\mapsto 0\) is a ring map; each case puts two free vectors in a line or three in a plane.
10.2 Four-dimensional extensions
Let \(V\subseteq \mathbb {C}\) be a \(\overline{\mathbb {Q}}\)-vector subspace of dimension at most \(4\), and let \(x_1,x_2\) and \(y_1,y_2,y_3\) be linearly independent over \(\overline{\mathbb {Q}}\) with all six products \(x_iy_j\) in \(V\). Then
for some \(b\neq 0\) and \(h\notin \overline{\mathbb {Q}}\).
Source: The converse direction, that such a progression carries a configuration, is used in [ Fis01 , Lemma 6.1 ] and [ Dia07 , Théorème 7(2) ] . This direction was not found in the sources read.
Put \(h=x_2/x_1\), which is not in \(\overline{\mathbb {Q}}\) and so is transcendental, and \(B=x_1(\overline{\mathbb {Q}}y_1+\overline{\mathbb {Q}}y_2+\overline{\mathbb {Q}}y_3)\). Both \(B\) and \(hB\) lie in \(V\), so \(W=B\cap hB\) has dimension at least \(2\). The spaces \(W\) and \(h^{-1}W\) lie in \(B\) and have dimension at least \(2\), so they meet in some \(c=hb\neq 0\) with \(b\in B\). Then \(b,hb,h^{2}b\in B\) and \(h^{3}b\in hB\): four free vectors in \(V\).
Let \(K\) be a field and \(P_0,P_1,P_2\in K[X]\) with \(\deg P_1,\deg P_2\le 3\), \(P_1P_2=P_0^{2}\), and \(P_0,P_1\) linearly independent over \(K\). Then there are \(Q_0,Q_1\in K[X]\) of degree at most \(1\), linearly independent over \(K\), and \(a,b\in K\) such that, with \(g=aQ_0+bQ_1\),
Source: Elementary.
Divide out a common factor: \(P_0=dQ_0\) and \(P_1=dQ_1\) with \(Q_0,Q_1\) coprime. Then \(Q_1P_2=dQ_0^{2}\), so \(Q_1\) divides \(d\), say \(d=gQ_1\). Degrees add, so \(\deg Q_i\le 1\). A relation between \(Q_0\) and \(Q_1\) would be one between \(P_0\) and \(P_1\), so they are independent; one of them has degree \(1\), hence \(\deg g\le 1\), and by Cramer’s rule \(g=aQ_0+bQ_1\).
Let \(u\in \mathbb {C}\setminus \overline{\mathbb {Q}}\) with \(u\bar u\) algebraic, let \(b\neq 0\) and \(h\notin \overline{\mathbb {Q}}\), and suppose that \(V=\overline{\mathbb {Q}}\, b+\overline{\mathbb {Q}}\, bh+\overline{\mathbb {Q}}\, bh^{2}+\overline{\mathbb {Q}}\, bh^{3}\) contains \(1\), \(u\) and \(\bar u\). Then \(V\) contains \(u^{2}\), or \(\bar u^{2}\), or \(1/(u-a)\) for some \(a\in \overline{\mathbb {Q}}\), \(a\neq 0\).
Source: A step of Theorem 10.6.
Write \(1=bP_0(h)\), \(u=bP_1(h)\) and \(u^{-1}=\bar u/(u\bar u)=bP_2(h)\) with \(\deg P_i\le 3\). As \(h\) is transcendental, \(P_1P_2=P_0^{2}\), and \(P_0,P_1\) are independent because \(u\notin \overline{\mathbb {Q}}\). Lemma 10.3 gives \(g=aQ_0+cQ_1\) and \(u=Q_1(h)/Q_0(h)\). If \(c=0\), then \(u^{2}=b\, aQ_1(h)^{3}\); if \(a=0\), then \(\bar u^{2}=b\, \rho ^{2}c\, Q_0(h)^{3}\) with \(\rho =u\bar u\); otherwise \(1/(u+a/c)=b\, c\, Q_0(h)^{2}Q_1(h)\).
Let \(u\in \mathbb {C}\setminus \overline{\mathbb {Q}}\) with \(u\bar u\) algebraic, and let \(w\) be \(u^{2}\), \(\bar u^{2}\), or \(1/(u-a)\) with \(a\in \overline{\mathbb {Q}}\), \(a\neq 0\). Then there are \(x_1,x_2\), linearly independent over \(\overline{\mathbb {Q}}\), and \(y_1,y_2,y_3\), linearly independent over \(\overline{\mathbb {Q}}\), with all six products \(x_iy_j\) in \(\overline{\mathbb {Q}}+\overline{\mathbb {Q}}u+\overline{\mathbb {Q}}\bar u+\overline{\mathbb {Q}}w\).
Source: The configurations behind [ Dia07 , Corollaire 5(1) and 5(4) ] , used there with Roy’s strong six exponentials theorem [ Roy92 ] .
Here \(u^{-1}=\bar u/(u\bar u)\) lies in the space. Take \((1,u)\otimes (u^{-1},1,u)\) for \(u^{2}\), \((1,u^{-1})\otimes (u,1,u^{-1})\) for \(\bar u^{2}\), and \((u^{-1},(u-a)^{-1})\otimes (1,u,u^{2}-au)\) for \(1/(u-a)\). The families are free because \(u\) is transcendental; the last one uses \(a\neq 0\).
Let \(u\in \mathbb {C}\setminus \overline{\mathbb {Q}}\) with \(u\bar u\) algebraic, put \(H_0=\overline{\mathbb {Q}}+\overline{\mathbb {Q}}u+\overline{\mathbb {Q}}\bar u\), and let \(z\notin H_0\). There are \(x_1,x_2\), linearly independent over \(\overline{\mathbb {Q}}\), and \(y_1,y_2,y_3\), linearly independent over \(\overline{\mathbb {Q}}\), with all six products \(x_iy_j\) in \(H_0+\overline{\mathbb {Q}}z\) if and only if \(z\in H_0+\overline{\mathbb {Q}}w\) for \(w=u^{2}\), \(w=\bar u^{2}\), or \(w=1/(u-a)\) with \(a\in \overline{\mathbb {Q}}\), \(a\neq 0\).
Source: Not found in the sources read, and not routine (Chapter 14). With Roy’s strong six exponentials theorem [ Roy92 ] , the “if” direction gives at a candidate the exclusions of [ Dia07 , Corollaire 5(1) and 5(4) ] ; every configuration has the shape of [ Fis01 , Lemma 6.1 ] and [ Dia07 , Théorème 7(2) ] .
If \(z\in H_0+\overline{\mathbb {Q}}w\) and \(z\notin H_0\), then \(H_0+\overline{\mathbb {Q}}z=H_0+\overline{\mathbb {Q}}w\), which carries a configuration by Proposition 10.5. Conversely, Lemma 10.2 writes \(H_0+\overline{\mathbb {Q}}z\) as \(b(\overline{\mathbb {Q}}+\overline{\mathbb {Q}}h+\overline{\mathbb {Q}}h^{2}+\overline{\mathbb {Q}}h^{3})\) with \(h\notin \overline{\mathbb {Q}}\), and Lemma 10.4 puts \(u^{2}\), \(\bar u^{2}\) or some \(1/(u-a)\) in it. That element is not in \(H_0\), since \(u\) is transcendental, so \(z\in H_0+\overline{\mathbb {Q}}w\).
Since \(\widetilde{\mathcal L}\), the \(\overline{\mathbb {Q}}\)-span of \(1\) and the logarithms of algebraic numbers, is closed under complex conjugation, a hypothesis \(z\in \widetilde{\mathcal L}\) also brings \(\bar z\); the five-dimensional spaces \(H_0+\overline{\mathbb {Q}}z+\overline{\mathbb {Q}}\bar z\) are the subject of the next section.
10.3 Five-dimensional extensions
Let \(u\in \mathbb {C}\setminus \overline{\mathbb {Q}}\) with \(\rho =u\bar u\) algebraic, let \(a\in \overline{\mathbb {Q}}\), \(a\neq 0\), and let \(z=u/(u^{2}-a)\) if \(a\bar a\neq \rho ^{2}\), or \(z=u/(u^{2}-a)^{2}\) if \(a\bar a=\rho ^{2}\). Then there are \(x_1,x_2\), linearly independent over \(\overline{\mathbb {Q}}\), and \(y_1,y_2,y_3\), linearly independent over \(\overline{\mathbb {Q}}\), with all six products \(x_iy_j\) in \(\overline{\mathbb {Q}}+\overline{\mathbb {Q}}u+\overline{\mathbb {Q}}\bar u+\overline{\mathbb {Q}}z+\overline{\mathbb {Q}}\bar z\).
Source: Rescaled, the configuration is the one in the proofs of [ Dia04 , Théorème 2 ] and [ Dia07 , Théorèmes 6(1) and 7(1) ] , and the progression of [ Fis01 , Lemma 6.1 ] .
In the first case put \(a'=\rho ^{2}/\bar a\), which differs from \(a\); then \(\bar z=-(\rho /\bar a)\, u/(u^{2}-a')\), and let \(b=1/(u(u^{2}-a)(u^{2}-a'))\). In the second, \(\bar z=(\rho /\bar a^{2})\, u^{3}/(u^{2}-a)^{2}\), and let \(b=1/(u(u^{2}-a)^{2})\). Partial fractions in \(u^{2}\), with \(u^{-1}=\bar u/\rho \), put \(b,bu^{2},bu^{4},bu^{6}\) in the space, so \((1,u^{2})\otimes (b,bu^{2},bu^{4})\) is a configuration, free because \(u\) is transcendental.
With \(u\), \(a\) and \(z\) as in Proposition 10.7, the space \(\overline{\mathbb {Q}}+\overline{\mathbb {Q}}u+\overline{\mathbb {Q}}\bar u+\overline{\mathbb {Q}}z+\overline{\mathbb {Q}}\bar z\) contains neither \(u^{2}\) nor \(\bar u^{2}\), nor \(1/(u-b)\) for any \(b\in \overline{\mathbb {Q}}\), \(b\neq 0\).
Source: Elementary.
Clearing denominators turns a membership into \(A(u^{2})+u\, B(u^{2})=0\) with \(A,B\in \overline{\mathbb {Q}}[X]\). The same identity holds at \(-u\), so \(A=B=0\). For \(u^{2}\) the odd part is not zero; for \(\bar u^{2}=\rho ^{2}/u^{2}\) the even part is not zero at \(0\); for \(1/(u-b)=(u+b)/(u^{2}-b^{2})\) the odd part is not zero at \(b^{2}\).
With \(u\), \(a\) and \(z\) as in Proposition 10.7, put \(H_0=\overline{\mathbb {Q}}+\overline{\mathbb {Q}}u+\overline{\mathbb {Q}}\bar u\) and \(W=H_0+\overline{\mathbb {Q}}z+\overline{\mathbb {Q}}\bar z\). Then \(W\) carries a configuration, but for no \(w\in W\setminus H_0\) does \(H_0+\overline{\mathbb {Q}}w\) carry one.
Source: Not found in the sources read, but short (Chapter 14). The shape of the configuration is printed; see Proposition 10.7.
The configuration is Proposition 10.7. If \(H_0+\overline{\mathbb {Q}}w\) carried one, Theorem 10.6 would give \(w\in H_0+\overline{\mathbb {Q}}t\) with \(t\) one of \(u^{2}\), \(\bar u^{2}\), \(1/(u-b)\). Since \(w\notin H_0\), exchange puts \(t\) in \(H_0+\overline{\mathbb {Q}}w\subseteq W\), against Lemma 10.8. The configuration does not need all of \(W\): its six products span the odd part \(\overline{\mathbb {Q}}u+\overline{\mathbb {Q}}\bar u+\overline{\mathbb {Q}}z+\overline{\mathbb {Q}}\bar z\).
Let \(u\in \mathbb {C}\setminus \overline{\mathbb {Q}}\) with \(\rho =u\bar u\) algebraic, and let \(a\in \overline{\mathbb {Q}}\), \(a\neq 0\), with \(a\bar a=\rho ^{2}\). Then there are no \(x_1,x_2\), linearly independent over \(\overline{\mathbb {Q}}\), and \(y_1,y_2,y_3\), linearly independent over \(\overline{\mathbb {Q}}\), with all six products \(x_iy_j\) in \(\overline{\mathbb {Q}}+\overline{\mathbb {Q}}u+\overline{\mathbb {Q}}\bar u+\overline{\mathbb {Q}}\, u/(u^{2}-a)\).
Source: A consequence of Theorem 10.9. It is where the hypotheses of [ Dia04 , Théorème 2 ] and of [ Dia07 , Corollaire 4(4) and Théorème 7(1) ] fail for this family.
Here \(z=u/(u^{2}-a)\) has \(\bar z=-(\rho /\bar a)z\), so the space is stable under conjugation. With \(z_2=u/(u^{2}-a)^{2}\), the element \(u/(u^{2}-a)=(\bar a^{2}/\rho )\bar z_2-az_2\) lies in \(H_0+\overline{\mathbb {Q}}z_2+\overline{\mathbb {Q}}\bar z_2\) and not in \(H_0\), so Theorem 10.9 applies. When \(a\bar a\neq \rho ^{2}\), adding \(\bar z\) gives a space that does carry a configuration.
If \(\lambda \) is a logarithm of an algebraic number, so is \(\bar\lambda \).
Source: Elementary.
\(e^{\bar\lambda }=\overline{e^{\lambda }}\), and the conjugate of an algebraic number is algebraic.
The \(\overline{\mathbb {Q}}\)-vector space \(\widetilde{\mathcal L}\) spanned by \(1\) and the logarithms of algebraic numbers is closed under complex conjugation.
Source: Elementary; it is the stability that [ Dia04 , Théorème 2 ] uses.
Conjugation fixes \(1\), sends logarithms to logarithms (Lemma 10.11), and \(\overline{c\lambda }=\bar c\, \bar\lambda \) with \(\bar c\in \overline{\mathbb {Q}}\).
Assume Roy’s strong six exponentials theorem, and let \(u\) be a candidate: \(u\neq 0\), \(|u|\) algebraic and \(e^{u}\) algebraic. Let \(a\in \overline{\mathbb {Q}}\), \(a\neq 0\), and \(\rho =u\bar u\). Then \(u/(u^{2}-a)\notin \widetilde{\mathcal L}\) if \(a\bar a\neq \rho ^{2}\), and \(u/(u^{2}-a)^{2}\notin \widetilde{\mathcal L}\) if \(a\bar a=\rho ^{2}\).
Source: One substitution in [ Dia04 , Théorème 2 ] , at \(x=(u,\bar u)\) and \(y=1/(u^{2}-a)\), and in [ Dia07 , Corollaire 4(4) and Théorème 7(1) ] ; the second part is [ Dia07 , Théorème 6(3) ] . Roy’s theorem is [ Roy92 ] .
\(\widetilde{\mathcal L}\) is closed under conjugation and contains \(1\), \(u\) and \(\bar u\), so \(z\in \widetilde{\mathcal L}\) would put the space of Proposition 10.7 in \(\widetilde{\mathcal L}\), with its configuration, against Roy’s theorem.
10.4 Power hulls
Let \(u\in \mathbb {C}\setminus \overline{\mathbb {Q}}\) and \(k\ge 1\). There are \(x_1,x_2\), linearly independent over \(\overline{\mathbb {Q}}\), and \(y_1,y_2,y_3\), linearly independent over \(\overline{\mathbb {Q}}\), with all six products \(x_iy_j\) in \(\overline{\mathbb {Q}}u^{-k}+\overline{\mathbb {Q}}u^{-1}+\overline{\mathbb {Q}}+\overline{\mathbb {Q}}u+\overline{\mathbb {Q}}u^{k}\) if and only if \(k=2\) or \(k=3\).
Source: Not found in the sources read, and not routine (Chapter 14). For \(k=2,3\) the configurations are geometric progressions of four elements, which [ Fis01 , Lemma 6.1 ] and [ Dia07 , Théorème 7(2) ] exclude from \(\widetilde{\mathcal L}\) with Roy’s theorem; at a candidate they give [ Dia07 , Corollaire 5(1) and 5(2) ] . The direction “only for \(k=2,3\)” was not found; Diaz remarks that inside that theorem one cannot hope to go very far [ Dia07 , p. 390 ] .
For \(k=2\) take \((1,u)\otimes (u^{-1},1,u)\), and for \(k=3\) take \((u,u^{-1})\otimes (1,u^{-2},u^{2})\). Conversely, multiply by \(u^{k}\): the products become polynomials in \(u\) supported on \(\{ 0,k-1,k,k+1,2k\} \), and since \(u\) is transcendental a rank-one \(2\times 3\) matrix of them, with free rows and columns, has rows that differ by a fixed non-zero shift of the orders at \(0\). For \(k=1\) and for \(k\ge 4\) no shift keeps three exponents of that set inside it.
10.5 Separation and the normal form
Let \(K\subseteq F\) be subfields of \(\mathbb {C}\), \(V_0\subseteq F\) a \(K\)-space and \(w\) transcendental over \(F\). If \(x_1,x_2\) and \(y_1,y_2\) are each linearly independent over \(K\) and every \(x_iy_j\) lies in \(V_0+Kw\), then every \(x_iy_j\) lies in \(V_0\).
Source: Not found in the sources read; short (Chapter 14).
Write \(x_iy_j=c_{ij}+\ell _{ij}w\) with \(c_{ij}\in V_0\), \(\ell _{ij}\in K\). Since \(w\) is transcendental over \(F\) and \(\det (C+wL)=0\), \(\det L=0\). A non-zero column \(j\) of \(L\) gives \(z=\ell _{2j}x_1-\ell _{1j}x_2\neq 0\) with \(zy_1,zy_2\in F\); comparing coefficients of \(w\) in \((zy_1)(x_iy_2)=(zy_2)(x_iy_1)\) gives \(z(\ell _{i2}y_1-\ell _{i1}y_2)=0\), so \(L=0\) by the independence of \(y\).
A \(p\times q\) configuration over \(K\) in a \(K\)-space \(V\subseteq \mathbb {C}\) is a pair of families \(x_1,\dots ,x_p\) and \(y_1,\dots ,y_q\), each linearly independent over \(K\), with every product \(x_iy_j\) in \(V\).
Let \(K\subseteq F\) be subfields of \(\mathbb {C}\), \(V_0\subseteq F\) a \(K\)-space and \(w_1,\dots ,w_m\) algebraically independent over \(F\). Every \(p\times q\) configuration over \(K\) with \(p,q\ge 2\) in \(V_0+Kw_1+\dots +Kw_m\) lies in \(V_0\).
Source: Not found in the sources read, and not routine (Chapter 14). It contains the separation steps of Theorems 10.1 and 12.3, which are over \(\overline{\mathbb {Q}}\).
Induction on \(m\), over \(F(w_1,\dots ,w_{m-1})\): every entry lies in a \(2\times 2\) block, to which Lemma 10.15 applies.
Let \(K\) be a field and \(X\subseteq K[X]\) a finite-dimensional \(K\)-space. The orders at \(0\) of the non-zero elements of \(X\) take exactly \(\dim X\) values.
Source: Standard: the valuation argument of [ BSZ17 , proof of Theorem 33 ] .
Induction on \(\dim X\): if \(n_0\) is the least order, attained by \(f_0\), the elements with coefficient \(0\) at \(X^{n_0}\) form a complement of \(Kf_0\) whose orders are the others.
For non-zero finite-dimensional \(K\)-spaces \(X,Y\subseteq K[X]\), the span \(XY\) of the products satisfies \(\dim X+\dim Y\le \dim XY+1\).
Source: A case of the linear Cauchy–Davenport theorem of Eliahou and Lecouvey, as stated in [ BSZ17 , Theorem 2 ] .
The order sets \(A,B\) of \(X,Y\) have \(\dim X\) and \(\dim Y\) elements (Lemma 10.17), and \(A+B\) lies in the order set of \(XY\); Cauchy–Davenport in \(\mathbb N\).
Let \(u\) be transcendental over a subfield \(K\subseteq \mathbb {C}\) and \(V_0\subseteq K(u)\) a finite-dimensional \(K\)-space. Every \(p\times q\) configuration over \(K\) in \(V_0\) has \(p+q\le \dim V_0+1\).
Source: Not found in the sources read; short (Chapter 14).
Clear denominators: \(g(u)x_iy_j=A_{ij}(u)\) with \(A_{i1}A_{1j}=A_{11}A_{ij}\) in \(K[X]\); the spans of the \(A_{i1}\) and of the \(A_{1j}\) have dimensions \(p\) and \(q\), and Lemma 10.18 bounds the span of their products, which embeds in \(V_0\).
Let \(u\) be transcendental over a subfield \(K\subseteq \mathbb {C}\) with \(\rho =u\bar u\in K\). Every \(p\times q\) configuration over \(K\) in \(K+Ku+K\bar u\) has \(p+q\le 4\).
Source: Not found in the sources read; short (Chapter 14). Over \(\overline{\mathbb {Q}}\) it is a case of Roy’s lemma [ Wal00 , Lemma 12.16 ] .
\(K+Ku+K\bar u\subseteq K(u)\) has dimension \(3\); Proposition 10.19.
Let \(u,w_1,\dots ,w_m\) be algebraically independent over a subfield \(K\subseteq \mathbb {C}\), with \(\rho =u\bar u\in K\). Every \(p\times q\) configuration over \(K\) with \(p,q\ge 2\) in \(K+Ku+K\bar u+Kw_1+\dots +Kw_m\) is \(2\times 2\) and lies in \(K+Ku+K\bar u\).
Source: Not found in the sources read; short (Chapter 14). For \(K=\overline{\mathbb {Q}}\) it contains Theorem 10.1.
Let \(u\) be transcendental over a subfield \(K\subseteq \mathbb {C}\) with \(\rho =u\bar u\in K\). Then \(x_1,x_2\) and \(y_1,y_2\) form a \(2\times 2\) configuration over \(K\) in \(K+Ku+K\bar u\) if and only if \(x_i=\mu (P_{i1}+P_{i2}u)\) and \(y_j=\mu ^{-1}(Q_{1j}+Q_{2j}\bar u)\) for some \(\mu \neq 0\) and invertible \(P,Q\in K^{2\times 2}\): the configurations form the orbit of \((1,u)\otimes (1,\bar u)\) under \(\mathrm{GL}_2(K)\times \mathrm{GL}_2(K)\).
Source: Not found in the sources read, and not routine (Chapter 14).
\(ux_iy_j=A_{ij}(u)\) with \(\deg A_{ij}\le 2\), and the vanishing minor factors \(A_{ij}=a_ib_j\) in \(K[X]\) (extract a gcd). Neither the \(a_i\) nor the \(b_j\) are both constant, so all have degree at most \(1\); then \(x_i=\mu a_i(u)\) and \(y_j=\mu ^{-1}b_j(u)/u\), with \(1/u=\bar u/\rho \).
In the setting of Corollary 10.21, for every \(2\times 2\) configuration there is an invertible \(c\in K^{2\times 2}\) with \(x_iy_j-c_{ij}\in Ku+K\bar u\). In particular no \(2\times 2\) configuration lies in \(Ku+K\bar u+Kw_1+\dots +Kw_m\).
Source: Not found in the sources read; short (Chapter 14). It sharpens Diaz.generic_qbar_homogeneous_four_exp_barrier, the homogeneous barrier of the companion note.
Let \(x_1,x_2\) and \(y_1,y_2\) be pairs, each linearly independent over a subfield \(K\subseteq \mathbb {C}\), whose four products span a \(K\)-space of dimension \(3\). Then \(\operatorname {span}(x_1,x_2)=Ka+Kah\) and \(\operatorname {span}(y_1,y_2)=Kb+Kbh\) for some \(h\notin K\) and \(a,b\neq 0\).
Source: The dimension-2 case of [ BSZ17 , Lemmas 4–5 ] , stated there for a base field algebraically closed in the extension; here over any subfield.
A relation \(x_1u'+x_2v=0\) with \(u',v\in \operatorname {span}(y)\), \(v\neq 0\); take \(h=x_2/x_1\), \(a=x_1\), \(b=v\), so that \(bh=-u'\).
An algebraic number in the \(\overline{\mathbb {Q}}\)-span of the logarithms of algebraic numbers is \(0\).
Source: A consequence of Baker’s theorem [ Bak66 ] .
Choose a \(\mathbb {Q}\)-linearly independent family of logarithms with the same \(\mathbb {Q}\)-span, write \(z\) on it with algebraic coefficients, and apply Theorem 9.24 with constant term \(-z\).
Let \(u\) be a candidate with \(u,w_1,\dots ,w_m\) algebraically independent over \(\overline{\mathbb {Q}}\). In every \(2\times 2\) configuration over \(\overline{\mathbb {Q}}\) in \(\overline{\mathbb {Q}}+\overline{\mathbb {Q}}u+\overline{\mathbb {Q}}\bar u+\overline{\mathbb {Q}}w_1+\dots +\overline{\mathbb {Q}}w_m\), each row and each column has an entry outside \(\overline{\mathbb {Q}}\mathcal L\).
Source: Not found in the sources read; short (Chapter 14).
10.6 Laurent hulls
A finite \(S\subseteq \mathbb {Z}\) contains \(A+B\) with \(|A|=2\), \(|B|=q\) if and only if some \(d\gt 0\) has \(\# \{ s\in S:s+d\in S\} \ge q\).
Source: Elementary.
For \(A=\{ a,a+d\} \), \(A+B\subseteq S\) says that \(a+B\) lies in \(\{ s\in S:s+d\in S\} \).
A \(p\times q\) configuration over \(K\) in a \(K\)-space \(V\subseteq \mathbb {C}\) is a pair of families \(x_1,\dots ,x_p\) and \(y_1,\dots ,y_q\), each linearly independent over \(K\), with every product \(x_iy_j\) in \(V\).
Let \(u\) be transcendental over a subfield \(K\subseteq \mathbb {C}\) and \(S\subseteq \mathbb {Z}\) finite. The \(K\)-span of the \(u^s\), \(s\in S\), carries a \(p\times q\) configuration (\(p,q\ge 1\)) if and only if \(A+B\subseteq S\) for some \(|A|=p\), \(|B|=q\).
Source: Not found in the sources read, and not routine (Chapter 14). Theorem 10.14 is the case \(S=\{ 0,\pm 1,\pm k\} \); the nearest printed result is [ Fis01 , Lemma 6.1 ] .
Distinct powers of \(u\) are independent. Conversely, \(P_{ij}(u)=u^Nx_iy_j\) with \(P_{ij}\) supported on \(S+N\); the spans of the \(P_{i1}\) and the \(P_{1j}\) have \(p\) and \(q\) orders at \(0\) (Lemma 10.17), and \(fg=P_{11}h\) puts their sums, shifted, in \(S\).
For integers \(4\le k\lt l\), some difference occurs three times in \(\{ 0,\pm 1,\pm k,\pm l\} \) if and only if \(l\in \{ k+1,k+2,2k-1,2k,2k+1,3k\} \).
Source: Elementary.
The positive differences are \(1,k-1,k,k+1,l-1,l,l+1,l-k,l+k\) twice each and \(2,2k,2l\) once each; a third occurrence needs one of the six coincidences.
Let \(u\) be transcendental over a subfield \(K\subseteq \mathbb {C}\) and \(4\le k\lt l\). The span of the \(u^s\), \(s\in \{ 0,\pm 1,\pm k,\pm l\} \), carries a \(2\times 3\) configuration if and only if \(l\in \{ k+1,k+2,2k-1,2k,2k+1,3k\} \).
Source: Not found in the sources read; short (Chapter 14). The exclusions it gives at a candidate are substitutions in [ Dia07 ] (Corollary 10.32).
Let \(u\) be transcendental over a subfield \(K\subseteq \mathbb {C}\). The span of the \(u^s\), \(s\in \{ 0,\pm 1\} \cup \{ \pm 4^j:j\ge 1\} \), carries no \(2\times 3\) configuration. So at a candidate, Roy’s theorem used through Laurent hulls cannot exclude \(u^{4^j}\in \widetilde{\mathcal L}\) for all \(j\).
Source: Not found in the sources read; short (Chapter 14). Not in [ Dia07 ] (p. 390) nor [ Fis01 ] .
Reduce to a finite part. For a pair \(s\lt s+d\) with \(M=\max (|s|,|s+d|)\), either \(s=-M\), \(d=2M\), or \(|d-M|\le M/4\) and \(s\in \{ M-d,-M\} \); distinct non-zero absolute values differ by a factor at least \(4\), so no difference occurs three times.
Assume Roy’s strong six exponentials theorem and let \(u\) be a candidate. For \(4\le k\lt l\) with \(l\in \{ k+1,k+2,2k-1,2k,2k+1,3k\} \), \(u^k\) and \(u^l\) are not both in \(\widetilde{\mathcal L}\).
Source: Each case is one substitution in [ Dia07 ] : Corollaire 2(P)(1), Théorème 7(1), Corollaire 5(1)–(2), with \(\bar u^{k}=\rho ^{k}u^{-k}\).
By conjugation \(u^s\in \widetilde{\mathcal L}\) for \(s\in \{ 0,\pm 1,\pm k,\pm l\} \); in each case an explicit \((u^a)_{a\in A}\otimes (u^b)_{b\in B}\) with \(A+B\subseteq \{ 0,\pm 1,\pm k,\pm l\} \) contradicts Roy’s theorem.
10.7 Two poles
Let \(u\in \mathbb {C}\setminus \overline{\mathbb {Q}}\) with \(\rho =u\bar u\) algebraic, \(a_1\neq a_2\) non-zero algebraic numbers, \(z_i=u/(u^{2}-a_i)\) and \(W=\overline{\mathbb {Q}}+\overline{\mathbb {Q}}u+\overline{\mathbb {Q}}\bar u+\overline{\mathbb {Q}}z_1+\overline{\mathbb {Q}}z_2\). Then \(W\) carries a \(2\times 3\) configuration: \(x=(1,u^{2})\), \(y=(b,bu^{2},bu^{4})\) with \(b=1/(u(u^{2}-a_1)(u^{2}-a_2))\).
Source: The progression of [ Fis01 , Lemma 6.1 ] and [ Dia07 , Théorème 7(2) ] , with ratio \(u^{2}\).
Partial fractions: \(bu^{2}=(z_1-z_2)/(a_1-a_2)\), \(bu^{4}=(a_1z_1-a_2z_2)/(a_1-a_2)\), \(bu^{6}=u+(a_1^{2}z_1-a_2^{2}z_2)/(a_1-a_2)\), and \(b\) is a combination of \(u^{-1}=\bar u/\rho \), \(z_1\), \(z_2\).
Let \(u\in \mathbb {C}\setminus \overline{\mathbb {Q}}\) with \(\rho =u\bar u\) algebraic, \(a_1\neq a_2\) non-zero algebraic numbers, \(z_i=u/(u^{2}-a_i)\) and \(W=\overline{\mathbb {Q}}+\overline{\mathbb {Q}}u+\overline{\mathbb {Q}}\bar u+\overline{\mathbb {Q}}z_1+\overline{\mathbb {Q}}z_2\). Then \(W\) contains none of \(u^{2}\), \(\bar u^{2}\) and \(1/(u-c)\) with \(c\in \overline{\mathbb {Q}}\) non-zero.
Source: Elementary.
Every element of \(W\) is a constant plus an odd function of \(u\); clearing denominators gives \(A(u^{2})+uB(u^{2})=0\), so \(A=B=0\), and each of the three numbers has a non-zero part of the wrong parity.
Let \(u\in \mathbb {C}\setminus \overline{\mathbb {Q}}\) with \(\rho =u\bar u\) algebraic, \(a_1\neq a_2\) non-zero algebraic numbers, \(z_i=u/(u^{2}-a_i)\) and \(W=\overline{\mathbb {Q}}+\overline{\mathbb {Q}}u+\overline{\mathbb {Q}}\bar u+\overline{\mathbb {Q}}z_1+\overline{\mathbb {Q}}z_2\). Then \(W\) carries a \(2\times 3\) configuration, but no \(\overline{\mathbb {Q}}+\overline{\mathbb {Q}}u+\overline{\mathbb {Q}}\bar u+\overline{\mathbb {Q}}w\) with \(w\in W\) outside \(\overline{\mathbb {Q}}+\overline{\mathbb {Q}}u+\overline{\mathbb {Q}}\bar u\) does.
Source: Not found in the sources read; short (Chapter 14). It drops the conjugation condition of Theorem 10.9.
Assume Roy’s strong six exponentials theorem and let \(u\) be a candidate. For distinct non-zero algebraic \(a_1,a_2\), the numbers \(u/(u^{2}-a_1)\) and \(u/(u^{2}-a_2)\) are not both in \(\widetilde{\mathcal L}\).
Source: One substitution in [ Dia07 , Théorème 7(2) ] , that is [ Fis01 , Lemma 6.1 ] .
Assume Roy’s strong six exponentials theorem, let \(u\) be a candidate and \(a_1\neq a_2\) algebraic with \(a_i\bar a_i=|u|^{4}\). Then \(u/(u^{2}-a_1)+u/(u^{2}-a_2)\notin \widetilde{\mathcal L}\).
Source: One line from a substitution in [ Dia07 , Théorème 7(2) ] .