Let \(\mathbb {K}\) be a nontrivially normed field, \(F\) a normed space over \(\mathbb {K}\), and \(g\colon \mathbb {K}^{\iota }\to F\) a function of class \(C^{k}\). For \(i\in \iota \) and a function \(h\) on \(\mathbb {K}^{\iota }\), let \(\partial _ih(z)\) be the derivative at \(w=z_i\) of \(w\mapsto h(z[i:=w])\), where \(z[i:=w]\) is \(z\) with its coordinate \(z_i\) replaced by \(w\). Then for every list \(L\) of \(k\) coordinate directions and every \(z\in \mathbb {K}^{\iota }\)

the partial derivative in the direction \(L(k-1)\) being taken first.

Source: Standard.

Let \(F\) be an entire function on \(\mathbb {C}^{\iota }\) with \(|F|\le M\) on the closed polydisc of radius \(\rho \gt 0\) about \(p\). For a list \(L\) of \(k\) coordinate directions, let \(\sigma _\nu \) be the number of times \(\nu \) occurs in \(L\). Then

Source: Classical; in the form of [ Wal00 , p. XVII ] .

Let \(K\) be a field of characteristic \(0\), \(E\subset K\) a finite set and \(S_0\in \mathbb {N}\), and write \(q^{(k)}\) for the \(k\)-th formal derivative of \(q\in K[X]\). There are polynomials \(b_{\zeta ,k}\in K[X]\) (\(\zeta \in E\), \(0\le k\lt S_0\)) of degree less than \(|E|S_0\) such that, for \(\zeta '\in E\) and \(0\le k'\lt S_0\), \(b_{\zeta ,k}^{(k')}(\zeta ')\) is \(1\) if \((\zeta ',k')=(\zeta ,k)\) and \(0\) otherwise, and every \(q\in K[X]\) of degree less than \(|E|S_0\) satisfies

Source: Standard (Hermite interpolation).

Let \(0\lt r\) and \(5r\le R\), let \(Z\) be a finite multiset of complex numbers of modulus at most \(r\), put \(p=|Z|\) and \(P(x)=\prod _{\zeta \in Z}(x-\zeta )\), and let \(u\) be an entire function with \(|u|\le M\) on the closed disc of radius \(R\) about \(0\). Then \(u=\rho +Pq\) for a polynomial \(\rho \) of degree less than \(p\) and an entire function \(q\), such that \(\rho ^{(k)}(\zeta )=u^{(k)}(\zeta )\) whenever \(k\) is less than the multiplicity of \(\zeta \) in \(Z\), and for \(|x|\le R\)

Source: [ Wal00 , Lemma 4.8 c)–d), p. 123 ] in one variable, with the bounds of Step 2.4 of its proof (pp. 125–126).

Let \(g\) be an entire function on \(\mathbb {C}^{\iota }\) with \(|g|\le K\) on the closed polydisc of radius \(R\) about \(0\), let \(i\in \iota \) and \(S_0\in \mathbb {N}\), and let \(E\) be a finite set of complex numbers of modulus at most \(r\), where \(0\lt r\) and \(5r\le R\); put \(p=|E|S_0\). There is an entire function \(h\) on \(\mathbb {C}^{\iota }\) such that:

• \(|h|\le 3^{p}K\) on the closed polydisc of radius \(R\) about \(0\);

• \(|g-h|\le (2r)^{p}(3/R)^{p}K\) on the closed polydisc of radius \(r\) about \(0\);

• for every point \(\xi \) and every list \(v\) of \(m\) coordinate directions different from \(i\): if \(D_{(v,i^{k})}\, g(\xi [i:=\zeta ])=0\) for every \(\zeta \in E\) and every \(k\lt S_0\), where \((v,i^{k})\) is the list \(v\) followed by \(k\) times \(i\), then \(D_vh(\xi )=0\).

Source: A step of the proof of [ Wal00 , Proposition 4.7, pp. 122–130 ] : the case \(m=n\) of its Lemma 4.8 (pp. 123–126) in one coordinate, with the bounds of Step 2.4.

Let \(J\) and \(\Lambda \) be finite sets, \(\mu =|J|\) and \(\nu =|\Lambda |\), and let \(v_{j\lambda }\in \mathbb {R}\) with \(\sum _\lambda |v_{j\lambda }|\le C\) for every \(j\), where \(C\gt 0\). Let \(X\ge 0\) and \(\ell \ge 1\) be integers with \(\ell ^{\mu }\lt (X+1)^{\nu }\). Then there is \(\xi \in \mathbb {Z}^{\Lambda }\), \(\xi \neq 0\), with \(|\xi _\lambda |\le X\) for every \(\lambda \) and

Source: [ Wal00 , Lemma 4.11, pp. 132–133 ] , with any real \(C\gt 0\) in place of an integer.

Let \(E\) be a complex normed space and \(G\colon E\to \mathbb {C}\) analytic at every point, with \(|G|\le M\) on the closed ball of radius \(R\) about \(0\), and let \(0\lt r\lt R\). For every \(z\in E\) with \(\| z\| \le r\) and every \(T\in \mathbb {N}\)

Source: The tail bound in the proof of [ Wal00 , Lemma 4.13, pp. 134–135 ] , with \(1/(1-r/R)\) in place of \(1+\sqrt{T}\).

Let \(\Lambda \) be a finite set, let \(\varphi _\lambda \) (\(\lambda \in \Lambda \)) be entire functions on \(\mathbb {C}^{\iota }\) with \(|\varphi _\lambda |\le B_\lambda \) on the closed polydisc of radius \(r\gt 0\) about \(0\), and let \(T\in \mathbb {N}\). There are \(u_{\tau \lambda }\in \mathbb {C}\), for \(\lambda \in \Lambda \) and \(\tau \in \{ 0,\dots ,T-1\} ^{\iota }\), with \(|u_{\tau \lambda }|\le B_\lambda \), such that for all \(c\in \mathbb {C}^{\Lambda }\) and \(z\in \mathbb {C}^{\iota }\) the function \(F=\sum _\lambda c_\lambda \varphi _\lambda \) satisfies

Source: A step of the proof of [ Wal00 , Proposition 4.10, p. 135 ] , with Cauchy’s inequalities (p. XVII).

Let \(\Lambda \) be a finite set, \(L=|\Lambda |\), and let \(\varphi _\lambda \) (\(\lambda \in \Lambda \)) be entire functions on \(\mathbb {C}^{\iota }\) with \(|\varphi _\lambda |\le B_\lambda \) on the closed polydisc of radius \(R\) about \(0\) and \(\sum _\lambda B_\lambda \le C\), where \(C\gt 0\). Let \(0\lt r\lt R\), and let \(T\), \(X\) and \(\ell \ge 1\) be integers with \(\ell ^{2T^{n}}\lt (X+1)^{L}\). Then there are integers \(p_\lambda \), not all zero, with \(|p_\lambda |\le X\), such that on the closed polydisc of radius \(r\) about \(0\)

Source: [ Wal00 , § 4.5: Lemmas 4.11–4.13 and the proof of Proposition 4.10, pp. 132–136 ] , with \(T\), \(X\) and \(\ell \) left free.

Let \(n\ge 1\) and \(L\) be integers and \(N,U,V,r,R\) real numbers with \(r\gt 0\), \(W=N+U+V\ge 12n^{2}\), \(er\le R\le re^{W/6}\) and

Then there are integers \(T\), \(X\) and \(\ell \ge 1\) with \(X\le e^{N}\) and \(\ell ^{2T^{n}}\lt (X+1)^{L}\) such that

Source: The numerical part of the proof of [ Wal00 , Proposition 4.10, pp. 135–136 ] .

Let \(K\) be a number field with an embedding \(\varphi \colon K\to \mathbb {C}\), let \(w\in K^{\iota }\), \(\tau \in \mathbb {N}\), \(k_0\in \iota \), and \(g(z)=z_{k_0}^{\tau }\exp \bigl(\sum _\nu \varphi (w_\nu )z_\nu \bigr)\). Let \(q\in \mathbb {C}^{\iota }\) and \(v\in K\), with \(\varphi (v)=q_{k_0}\) if \(\tau \gt 0\). Suppose that \(A\ge 1\), \(\overline{\left|w_\nu \right|}\le A\) for every \(\nu \), \(\overline{\left|v\right|}\le B\), and that \(\delta w_\nu \) and \(\delta v\) are algebraic integers for an integer \(\delta \). Then for every list \(L\) of \(k\) coordinate directions

for some \(\gamma \in K\) with \(\overline{\left|\gamma \right|}\le A^{k}(B+k)^{\tau }\) and \(\delta ^{k+\tau }\gamma \) an algebraic integer.

Source: [ Wal00 , Lemma 4.9, p. 130 ] , in the form used in its §4.6.

Let \(x_1,\dots ,x_{d_1}\in \mathbb {C}^{\iota }\) be linearly independent over \(\mathbb {Q}\), let \(k_0\in \iota \) and \(T_0,T_1\in \mathbb {N}\), and let the coefficients \(p_{\tau ,t}\in \mathbb {C}\) be not all zero. Then \(D^{k}F_p(0)\neq 0\) for some \(k\).

Source: [ Wal00 , Exercises 2.4–2.5, p. 60 ] , in the form used in its §4.6.

Let \(K\) be a number field with an embedding \(\varphi \colon K\to \mathbb {C}\), let \(k_0\in \iota \), and let \(x_1,\dots ,x_{d_1}\) and \(y_j\) (\(j\in \iota \)) be vectors in \(\mathbb {C}^{\iota }\). Suppose that \(x_{i\nu }=\varphi (\xi _{i\nu })\) and \(e^{\langle x_i,y_j\rangle }=\varphi (a_{ij})\) with \(\xi _{i\nu },a_{ij}\in K\), and that \(\eta _j\in K\) satisfy \(\varphi (\eta _j)=y_{jk_0}\) if \(T_0\gt 0\). Suppose that an integer \(\delta \) makes every \(\delta \xi _{i\nu }\), \(\delta \eta _j\) and \(\delta a_{ij}\) an algebraic integer, and that all their houses are at most \(H\ge 1\). Let \(T_1\ge 1\) and \(X\in \mathbb {R}\), and let the coefficients \(p_{\tau ,t}\) be integers with \(|p_{\tau ,t}|\le X\). Then at every grid point \(\sum _js_jy_j\) and for every list \(L\) of \(k\) coordinate directions, \(D_LF_p\bigl(\sum _js_jy_j\bigr)=\varphi (\gamma )\) for some \(\gamma \in K\) with

and \(\delta ^{k+T_0+d_1nT_1S_1}\gamma \) an algebraic integer.

Source: A step of [ Wal00 , § 4.6, step 2, p. 137 ] , with Lemma 4.9 (pp. 130–131).

Let \(k_0\in \iota \) and \(d_0\le 1\), and let \(x_1,\dots ,x_{d_1}\) and \(y_j\) (\(j\in \iota \)) be vectors in \(\mathbb {C}^{\iota }\) such that every coordinate \(x_{i\nu }\) and every \(e^{\langle x_i,y_j\rangle }\) is algebraic and, if \(d_0=1\), every \(y_{jk_0}\) is algebraic. There is \(C\ge 1\) with the following property. Let \(T\ge 1\) and \(S_1\) be integers and \(N\ge 0\), and let \(F_p\), with \(T_0=d_0T\) and \(T_1=T\), have integer coefficients with \(|p_{\tau ,t}|\le e^{N}\). If \(\Delta =D_LF_p\bigl(\sum _js_jy_j\bigr)\neq 0\) for a list \(L\) of \(k\) coordinate directions and a grid point \(\sum _js_jy_j\), then

Source: [ Wal00 , (4.14), § 4.6, step 2, p. 137 ] , with a bracket of a slightly different shape.

Let \(k_0\in \iota \), \(d_0\le 1\) and \(d=d_0+d_1\), and let \(x_1,\dots ,x_{d_1}\) and \(y_j\) (\(j\in \iota \)) be vectors in \(\mathbb {C}^{\iota }\) with \(\sum _{i,\nu }|x_{i\nu }|\le c_x\) and \(\sum _j\| y_j\| \le c_y\). Let \(S_1,T\ge 1\) be integers and \(U,N\gt 0\) and \(E\) real numbers; put \(r=(c_y+2)S_1\) and \(R=Er\), and assume

Then there are integers \(p_{\tau ,t}\), not all zero, with \(|p_{\tau ,t}|\le e^{N}\), such that, with \(T_0=d_0T\) and \(T_1=T\):

• \(\bigl|D_LF_p\bigl(\sum _js_jy_j\bigr)\bigr|\le k!\, e^{-U}\) at every grid point and for every list \(L\) of \(k\) coordinate directions;

• \(|F_p(w)|\le (T+1)^{d}e^{N}\rho ^{T}e^{c_xT\rho }\) whenever \(\rho \ge 1\) and \(\| w\| \le \rho \).

Source: [ Wal00 , § 4.6, step 3 and (4.16), pp. 138–139 ] .

Let \(n=|\iota |\ge 1\) and let \((y_j)_{j\in \iota }\) be a basis of \(\mathbb {C}^{\iota }\). There is \(c\ge 1\) with the following property. Let \(F\) be an entire function on \(\mathbb {C}^{\iota }\), let \(S_0\), \(S_1\ge 1\) and \(M\ge nS_0\) be integers, and let \(\rho \ge 1\) and \(B\) be real numbers. Suppose that every derivative of \(F\) of total order less than \(M\) vanishes at every grid point, and that \(|F(w)|\le B\) whenever \(\| w\| \le cS_1\rho \). Then at every grid point and for every list \(L\) of \(M\) coordinate directions

Source: [ Wal00 , § 4.6, step 5, p. 140 ] , with Proposition 4.7 (p. 122) and Cauchy’s inequalities (p. XVII).

Let \(n\ge 1\) be an integer and \(C\ge 1\). There is an integer \(s_0\) such that for every integer \(S_1\ge s_0\) there is \(e_0\) with the following property. Let \(T\ge 1\) and \(E\ge e_0\) be integers with \(\log T\le n\log S_1+n\log E\), and put

Then for every integer \(k\lt nET\)

Source: [ Wal00 , § 4.6, step 4 and (4.17), p. 139 ] .

Let \(n\ge 1\) and \(d\) be integers, \(C\ge 1\), \(c_x\in \mathbb {R}\) and \(c_F\ge 1\). There is an integer \(s_0\) such that for every integer \(S_1\ge s_0\) there is \(e_0\) with the following property. Let \(T\ge 1\) and \(E\ge e_0\) be integers with \(\log T\le n\log S_1+n\log E\), and let \(U\) and \(N\) be as in Lemma 9.19. Then for all integers \(u\ge ET\) and \(M\le 2nu\), with \(\rho '=c_FS_1u/T\),

Source: [ Wal00 , § 4.6, step 6 and (4.19), pp. 140–141 ] .

Let \(1\le n\lt d\) be integers, \(C\ge 1\), \(c_x,c_y\ge 0\) and \(c_F\ge 1\). There are integers \(S_1,T,E\ge 1\) and real numbers \(U,N\gt 0\) such that, with \(r=(c_y+2)S_1\) and \(R=Er\):

1. the four assumptions of Lemma 9.17 hold;

2. the inequality of Lemma 9.19 holds for every integer \(k\lt nET\);

3. the inequality of Lemma 9.20 holds for all integers \(u\ge ET\) and \(M\le 2nu\), with \(\rho '=c_FS_1u/T\).

Source: [ Wal00 , § 4.6: the conditions (4.15)–(4.19) and the choice of parameters in step 6, pp. 138–141 ] .