Let \(P\in \mathbb {Z}[X]\) be non-zero and let \(Q\in \mathbb {Z}[X]\) divide \(P\). If every coefficient of \(P\) has absolute value at most \(H\), then every coefficient of \(Q\) has absolute value at most \(e^{\deg P}H\).

Source: [ Wal71 , § 3, Lemme 1 ] (Gel’fond; a generalisation of a lemma of Popken and Koksma).

Let \(\alpha \in \mathbb {C}\) be transcendental and let \(P\in \mathbb {Z}[X]\) be primitive, with every coefficient of absolute value at most \(H\). Let \(n,\lambda \in \mathbb {R}\) with \(\deg P\le n\le \log H\) and \(\lambda \gt 6\). If \(|P(\alpha )|\lt H^{-\lambda n}\), then there are a primitive irreducible divisor \(Q\) of \(P\) and an integer \(s\ge 1\) with

and every coefficient of \(Q\) of absolute value at most \(H^{1/s}e^{2n/s}\).

Source: [ Wal71 , § 3, Lemme 2 ] , after Gel’fond [ Gel52 ] .

Let \(P,Q\in \mathbb {Z}[X]\) with \(Q\) irreducible, let \(\alpha \in \mathbb {C}\), and let \(H,h\ge 1\) bound the absolute values of the coefficients of \(P\) and of \(Q\) respectively. Put \(d=\deg P\) and \(\delta =\deg Q\). If

then \(Q\) divides \(P\).

Source: [ Wal71 , § 3, (3.13) ] , citing [ Lang66 , Ch. V, § 2 ] .

Let \(\alpha \in \mathbb {C}\) and \(\varepsilon \gt 0\). There is \(U\) such that for all real \(u\ge U\) and \(1\le v\le u\) the following holds. Let \(P,Q\in \mathbb {Z}[X]\) with \(Q\) irreducible, the coefficients of \(P\) of absolute value at most \(e^{u}\) and \(\deg P\le v\), the coefficients of \(Q\) of absolute value at most \(e^{3u}\) and \(\deg Q\le (1+\varepsilon /2)v\), and \(|P(\alpha )|\lt e^{-(4+\varepsilon )uv}\), \(|Q(\alpha )|\lt e^{-(4+\varepsilon )uv}\). Then \(Q\) divides \(P\).

Source: A step of the proof of [ Wal71 , § 3, Lemme fondamental ] .

Let \(\alpha \in \mathbb {C}\) and \(\varepsilon \gt 0\). Let \(\sigma _1,\sigma _2\colon \mathbb {R}\to \mathbb {R}\) be continuous, strictly increasing and tending to \(+\infty \), and let \(a_1,a_2\ge 1\) be such that for every \(x\ge 1\)

Put \(C=\max \{ 10+\varepsilon ,(4+\varepsilon )a_1a_2\} \). Suppose that for every integer \(N\gt N_0\) there is a non-zero \(P_N\in \mathbb {Z}[X]\) with every coefficient of absolute value at most \(e^{\sigma _1(N)}\), \(\deg P_N\le \sigma _2(N)\) and \(|P_N(\alpha )|\lt e^{-C\sigma _1(N)\sigma _2(N)}\). Then \(\alpha \) is algebraic.

Source: [ Wal71 , § 3, Lemme fondamental ] and its proof, where continuity of the \(\sigma _i\) is assumed without loss of generality.

Let \(w_1,\dots ,w_l\in \mathbb {C}\) be pairwise distinct with \(|w_j|\le W\) for some \(W\ge 0\). For each \(j\) let \(P_j\in \mathbb {C}[z]\) be zero or of degree less than \(q_j\), and put \(g(z)=\sum _jP_j(z)e^{w_jz}\) and \(n=\sum _jq_j\). Let \(R\ge 0\). If \(|g^{(s)}(0)|\le D\) for every \(s\lt n\), then

Source: [ Wal71 , § 4, (4.5)–(4.13) ] .

Let \(f(z)=\sum _{j=1}^{l}\sum _{i\lt q_j}b_{j,i}z^{i}e^{\omega _jz}\) with pairwise distinct \(\omega _j\) and \(b_{j,i}\) not all zero; put \(n=\sum _jq_j\) and \(\Omega =\max _j|\omega _j|\), and suppose \(\Omega \gt 0\). Let \(z_0\in \mathbb {C}\), \(\rho \ge 0\), let \(S\) be a finite set of points of the disc \(|z-z_0|\le \rho \), and put \(\sigma =\sum _{z\in S}\operatorname {ord}_zf\) and \(x=\rho \Omega \). Then for every \(R\gt x+1\)

Source: [ Wal71 , § 4, (4.14) ] and the rescaling that follows it.

Let \(n\ge 2\) be an integer and let \(x\ge 0\), \(\lambda \gt 0\) and \(\sigma \ge 0\) be real. If for every \(R\gt x+1\)

then

Source: After [ Wal71 , § 4, (4.14) ] .

For every integer \(n\ge 1\), real \(x\ge 0\) and \(\lambda \gt 0\), and every integer \(0\le \sigma \le n-1\),

where for \(n=1\) the second term is read as \(0\).

Source: Elementary.

Let \(f\), \(n\) and \(\Omega \) be as in Lemma 6.8, without the hypothesis \(\Omega \gt 0\), and suppose that \(n\le 1\) or \(\Omega =0\). Then for every finite set \(S\subset \mathbb {C}\)

Source: Elementary; the degenerate cases of [ Wal71 , § 4, Lemme 3 ] .

Let \(x_1,x_2\) be linearly independent over \(\mathbb {Q}\), and likewise \(y_1,y_2\). Let \(S,T,R_1,R_2,S'\in \mathbb {N}\) and let \(c(i,j,h)\in \mathbb {C}\) (\(i\lt S\), \(j,h\lt T\)) be not all zero; put

If for some \(\lambda \gt 0\)

then \(G^{(s)}(ay_1+by_2)\neq 0\) for some \(a\lt R_1\), \(b\lt R_2\) and \(s\lt S'\).

Source: [ Wal73 , Lemme 6 ] , with the zero estimate [ Wal71 , § 4, Lemme 3 ] in place of Gel’fond’s zero lemma.

Let \(l_{11},l_{12},l_{21},l_{22}\) be non-zero complex numbers with every \(e^{l_{ij}}\) algebraic, \(l_{11}l_{22}=l_{12}l_{21}\) and \(\operatorname {trdeg}_{\mathbb {Q}}\mathbb {Q}[l_{11},l_{12},l_{21},l_{22}]\le 1\), and suppose that neither the rows nor the columns of \((l_{ij})\) are linearly dependent over \(\mathbb {Q}\). Then there are \(x_1,x_2,y_1,y_2\in \mathbb {C}\), with \(x_1,x_2\) linearly independent over \(\mathbb {Q}\) and likewise \(y_1,y_2\), such that every \(e^{x_iy_j}\) is algebraic and \(\operatorname {trdeg}_{\mathbb {Q}}\mathbb {Q}[x_1,x_2,y_1,y_2]\le 1\).

Source: Elementary; as in [ Wal73 , § I ] .

Let \(x_1,x_2\) be linearly independent over \(\mathbb {Q}\), and likewise \(y_1,y_2\). If every \(e^{x_iy_j}\) is algebraic and \(\operatorname {trdeg}_{\mathbb {Q}}\mathbb {Q}[x_1,x_2,y_1,y_2]\le 1\), then \(x_1,x_2,y_1,y_2\) have presentation data.

Source: The opening of [ Wal73 , § III ] .

For all \(c,r\in \mathbb {N}\) there is \(\kappa \gt 0\) such that for all integers \(N\ge 3\) and \(S,T,a,b,m\in \mathbb {N}\) with

one has

Source: The parameter estimates in the proofs of [ Wal73 , Lemmes 4 and 7 ] .

Let \(x_1,x_2,y_1,y_2\in \mathbb {C}\) with every \(e^{x_iy_j}\) algebraic. Let \(\omega ,\omega _1\in \mathbb {C}\), let \(\varphi \) be evaluation at \((\omega ,\omega _1)\), and let \(D,E_i,G_j,H_{ij}\in \mathbb {Z}[X][Y]\) with \(\varphi (D)\neq 0\), \(x_i\varphi (D)=\varphi (E_i)\), \(y_j\varphi (D)=\varphi (G_j)\) and \(e^{x_iy_j}\varphi (D)=\varphi (H_{ij})\). There is \(c\in \mathbb {N}\) such that for all \(S,T,a,b,m\in \mathbb {N}\) there are \(\Lambda \in \mathbb {C}\), \(\Lambda \neq 0\), with \(|\Lambda |\le c^{\, c(1+m+S+T(a+b))}\), and \(P_{ijh}\in \mathbb {Z}[X][Y]\) (\(i\lt S\), \(j,h\lt T\)) of degree at most \(c(1+m+S)\) in \(X\) and in \(Y\) and of length at most

such that for all \(f_{ijh}\in \mathbb {C}\)

Source: A step of the proofs of [ Wal73 , Lemmes 4 and 7 ] .

In the setting of Lemma 6.17, let moreover \(Q\in \mathbb {Z}[X][Y]\) be monic in \(Y\) with \(\varphi (Q)=0\). There is \(c\in \mathbb {N}\) such that for all \(S,T,M,a,b,m\in \mathbb {N}\) there are \(\Lambda \neq 0\) with \(|\Lambda |\le c^{\, c(1+m+S+T(a+b))}\) and \(R_{ijh\mu \nu }\in \mathbb {Z}[X][Y]\) (\(i\lt S\), \(j,h\lt T\), \(\mu \lt M\), \(\nu \lt \deg _YQ\)) of \(Y\)-degree less than \(\deg _YQ\) and \(X\)-degree at most \(M+c(1+m+S)\), and of length at most

such that for every integer array \(q\)

where \(F_q\) is formed with \(T\) in place of \(2N\).

Source: A step of the proofs of [ Wal73 , Lemmes 4 and 7 ] .

Let \(x_1,x_2,y_1,y_2\in \mathbb {C}\) have presentation data, and suppose every \(e^{x_iy_j}\) is algebraic. There is \(\kappa _1\gt 0\) such that for every large \(N\) there are an integer \(M\) with \(0\lt M\le \kappa _1S\) and an \(R\times U\) integer matrix \(B\), where \(U=S(2N)^{2}Md\) and \(2R\le U\), with entries of absolute value at most \(e^{\kappa _1N^{2}\sqrt{\log N}}\), such that for every integer vector \(q\) with \(Bq=0\)

Source: The equation count of [ Wal73 , Lemme 4 ] .

Let \(\omega ,\omega _1\in \mathbb {C}\) and \(Q\in \mathbb {Z}[X][Y]\) with \(d=\deg _YQ\ge 1\), such that no non-zero \(A\in \mathbb {Z}[X][Y]\) with \(\deg _YA\lt d\) vanishes at \((\omega ,\omega _1)\), and let \(\kappa _1\gt 0\). There is \(\kappa \ge \kappa _1\) such that for every large \(N\), every \(M\) with \(0\lt M\le \kappa _1S\), and every \(R\times U\) integer matrix \(B\) with \(U=S(2N)^{2}Md\), \(2R\le U\) and entries of absolute value at most \(e^{\kappa _1N^{2}\sqrt{\log N}}\), there is an integer vector \(q\) with \(Bq=0\) such that every \(|q(i,j,h,\mu ,\nu )|\le e^{\kappa N^{2}\sqrt{\log N}}\), some \(c_q(i,j,h)\neq 0\), and every \(|c_q(i,j,h)|\le e^{\kappa N^{2}\sqrt{\log N}}\).

Source: Siegel’s lemma [ Lang66 , Ch. I, § 2 ] , as used in [ Wal73 , Lemme 4 ] .

Let \(x_1,x_2,y_1,y_2\in \mathbb {C}\) have presentation data, and suppose every \(e^{x_iy_j}\) is algebraic. There is \(\kappa \gt 0\) such that for every large \(N\) there are \(M\le \kappa S\) and integers \(q(i,j,h,\mu ,\nu )\) (\(i\lt S\), \(j,h\lt 2N\), \(\mu \lt M\), \(\nu \lt d\)) of absolute value at most \(e^{\kappa N^{2}\sqrt{\log N}}\), such that the coefficients \(c_q(i,j,h)\) are not all zero and of absolute value at most \(e^{\kappa N^{2}\sqrt{\log N}}\), and

Source: [ Wal73 , Lemme 4 ] , with Siegel’s lemma.

Let \(X\ge 0\), \(Y\ge 1\) and \(\kappa \ge 0\) be real, and let \(N,S,t_1,t_2,s\) be natural numbers with \(s\le S\) and \(N\ge 32(13+\kappa +\log Y+2XY)\). Write \(w=\sqrt{\log N}\) and suppose \(Sw\le N^{2}\le 2Sw\), \(N\le 2t_1w\) and \(Nw\le 2t_2\). Then

Source: The parameter estimate in the proof of [ Wal73 , Lemme 5 ] .

Let \(x_1,x_2\in \mathbb {C}\), let \(y_1,y_2\) be linearly independent over \(\mathbb {Q}\), and let \(\kappa \gt 0\). There are \(\kappa '\gt 0\) and \(N_0\) such that for every \(N\gt N_0\) and all coefficients with \(|c(i,j,h)|\le e^{\kappa N^{2}\sqrt{\log N}}\): if \(F^{(m)}(ay_1+by_2)=0\) for all \(a\lt t_1\), \(b\lt t_2\) and \(m\lt S\), then

Source: [ Wal73 , Lemme 5 ] .

Let \(\omega ,\omega _1\in \mathbb {C}\) and let \(Q\in \mathbb {Z}[X][Y]\) be monic in \(Y\) of degree \(d\), with \(Q(\omega ,\omega _1)=0\) and minimal there. There is \(c\in \mathbb {N}\) such that for every \(A\in \mathbb {Z}[X][Y]\) of \(Y\)-degree less than \(d\), length at most \(b\) and \(X\)-degree at most \(e\), with \(A(\omega ,\omega _1)\neq 0\), there is a non-zero \(P\in \mathbb {Z}[X]\) of length at most \((cb)^{d}\) and degree at most \(d(e+c)\) with

Source: The core of the proof of [ Wal73 , Lemme 7 ] .

Let \(x_1,x_2,y_1,y_2\in \mathbb {C}\) have presentation data, suppose every \(e^{x_iy_j}\) is algebraic, and let \(\kappa ,\kappa '\gt 0\). There is \(k\gt 0\) such that for every \(C\) there is \(N_0\) with the following property. Let \(N\gt N_0\), \(M\le \kappa S\), and let \(q(i,j,h,\mu ,\nu )\) be integers of absolute value at most \(e^{\kappa N^{2}\sqrt{\log N}}\). If \(a\lt R_1\), \(b\lt R_2\), \(s\lt S'\) and

then some \(P\in \mathbb {Z}[X]\) is small at \(\omega \) for \((N,C)\) with respect to the growth functions of constant \(k\).

Source: [ Wal73 , Lemme 7 ] .

Let \(x_1,x_2\) be linearly independent over \(\mathbb {Q}\), and likewise \(y_1,y_2\). Suppose every \(e^{x_iy_j}\) is algebraic and \(\operatorname {trdeg}_{\mathbb {Q}}\mathbb {Q}[x_1,x_2,y_1,y_2]\le 1\). Then there are a transcendental \(\omega \in \mathbb {C}\) and \(k\gt 0\) such that for every \(C\) and every large \(N\) there are coefficients \(c(i,j,h)\), not all zero, with the following property: whenever \(a\lt R_1\), \(b\lt R_2\), \(s\lt S'\) and \(F^{(s)}(ay_1+by_2)\neq 0\), some \(P\in \mathbb {Z}[X]\) is small at \(\omega \) for \((N,C)\) with respect to the growth functions of constant \(k\).

Source: [ Wal73 , § III, (4) and Lemmes 4, 5, 7 ] .

For every \(k\gt 0\) the growth functions \(\sigma _1,\sigma _2\) are strictly increasing and tend to \(+\infty \), and for every \(x\gt 0\)

Source: Elementary; the functions of the end of [ Wal73 , § III ] .

Let \(X,Y_1,Y_2\ge 0\). There is \(N_0\) such that for every integer \(N\gt N_0\), with \(T=2N\), \(\lambda =1/20\) and \(n=ST^{2}\),

Source: The parameters of [ Wal73 , (4) and Lemme 6 ] against the zero estimate [ Wal71 , § 4, Lemme 3 ] .

Let \(l_{11},l_{12},l_{21},l_{22}\) satisfy the hypotheses of Lemma 6.14. Then there are a transcendental \(\omega \), functions \(\sigma _1,\sigma _2\) and constants \(a_1,a_2\ge 1\) satisfying the hypotheses of Theorem 6.6, and pairs \(x_1,x_2\) and \(y_1,y_2\), each linearly independent over \(\mathbb {Q}\), with the following property. For every \(C\) and every large \(N\) there are natural numbers \(S,T,R_1,R_2,S'\), coefficients \(c(i,j,h)\in \mathbb {C}\) (\(i\lt S\), \(j,h\lt T\)), not all zero, and \(\lambda \gt 0\) such that

• the inequality of Lemma 6.13 holds, and

• whenever \(a\lt R_1\), \(b\lt R_2\), \(s\lt S'\) and \(G^{(s)}(ay_1+by_2)\neq 0\), where

  \[ G(z)=\sum _{i\lt S}\sum _{j,h\lt T}c(i,j,h)\, z^{i}e^{(jx_1+hx_2)z}, \]

  some \(P\in \mathbb {Z}[X]\) is small at \(\omega \) for \((N,C)\) with respect to \(\sigma _1,\sigma _2\).

Source: [ Wal73 , § III, (4) and Lemmes 4, 5, 7 ] .

Let \(l_{11},l_{12},l_{21},l_{22}\) be non-zero complex numbers with every \(e^{l_{ij}}\) algebraic, \(l_{11}l_{22}=l_{12}l_{21}\) and \(\operatorname {trdeg}_{\mathbb {Q}}\mathbb {Q}[l_{11},l_{12},l_{21},l_{22}]\le 1\), and suppose that neither the rows nor the columns of \((l_{ij})\) are linearly dependent over \(\mathbb {Q}\). Then there are a transcendental \(\omega \in \mathbb {C}\), functions \(\sigma _1,\sigma _2\) and constants \(a_1,a_2\ge 1\) satisfying the hypotheses of Theorem 6.6, such that for every \(C\in \mathbb {R}\) there is \(N_0\) and for every \(N\gt N_0\) a polynomial \(P_N\in \mathbb {Z}[X]\) that is small at \(\omega \) for \((N,C)\) with respect to \(\sigma _1,\sigma _2\).

Source: [ Wal73 , § III, Lemmes 4–7 ] , with the zero estimate of [ Wal71 , § 4, Lemme 3 ] ; see also [ Bro74 ] .