= General lower bound for a linear form in logarithms
{title2=$\Lambda=\beta_0+\sum_j\beta_j\log\alpha_j$}
Choose logarithms of nonzero <algebraic numbers> $\alpha_1,\ldots,\alpha_n$ and <algebraic numbers> $\beta_0,\ldots,\beta_n$, and put
$$
\Lambda=\beta_0+\sum_{j=1}^n\beta_j\log\alpha_j.
$$
Let $H(\gamma)$ be the <naive polynomial height> of the primitive <algebraic minimal polynomial> of $\gamma$. Choose $A_j\ge\max\{10,H(\alpha_j),\exp(|\log\alpha_j|)\}$ and $B\ge\max\{10,H(\beta_0),\ldots,H(\beta_n),\log A_1,\ldots,\log A_n\}$. There is an effectively computable $C>0$, depending only on $n$ and the <field extension degree> of the <number field> generated by the data, such that
$$
\Lambda\ne0\quad\Longrightarrow\quad |\Lambda|>\exp\!\left(-C\prod_{j=1}^n\log A_j\,\log B\right).
$$
The algebraic constant term is included here. Theorem 1.1 of the primary Philippon–Waldschmidt paper bounds $|\Lambda|$ below by $\exp[-C_0(n,D)\Omega(\log B+\log\log A)]$, with $\Omega=\prod_j\log A_j$ and $A=\max\{A_j,e^e\}$. Our assumption $B\ge\log A_j$ absorbs the extra term into $\log B$. Increasing each $A_j$ by a fixed power depending only on $n$ meets their normalization $A_j\ge e^n$, with the change absorbed into $C$. Thus this is a corollary of their general algebraic-coefficient theorem, including $\beta_0$, rather than an application of only the homogeneous <integer>-coefficient bound.<Lower bounds for linear forms in logarithms by Patrice Philippon and Michel Waldschmidt>[Theorem 1.1, printed pages 280–281; proof following Theorem 2.1]{ref}
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