Let $K$ be a number field, and let $S$ be a finite collection of places of $K$, including all the infinite places. We say $x$ is an $S$-unit if $v(x)=0$ for all $x \not \in S$. Let $O_S$ be the set of all $S$-units.
An $S$-unit equation is an equation of the form $ax+by=1$ with $a, b$ given, $x,y \in O_S$ and $x$ and $y$ coprime to each other. It is well known that if $a,b$ and $S$ are given, then we can effectively solve the $S$-unit equation $ax+by=1$, in the sense that we can find an explicit bound using Baker's effective linear forms of logarithms. In fact, with the work of Smart, Wildanger, et. al., it seems possible to solve most reasonable $S$-unit equations (i.e. when $S$ is not too large.)
I will try to put the details of how this can be done (as far as I understand it) in this post. To simplify the exposition, we will work over rationals, and we let $a=b=1$. (I think generalizing to other cases is clear.) In this case, given $S=\{p_0=\infty, p_1,p_2,\ldots,p_k\}$, we are asked to find $\alpha=(\alpha_1,\ldots,\alpha_k)^t$ and $\beta=(\beta_1,\ldots,\beta_k)^t$ so that
\[ \prod_{i=1}^k p_i^{\alpha_i} + \prod_{i=1}^k p_i^{\beta_i} = 1. \]
Therefore $v_i(x)=\alpha_i$ and $v_i(y)=\beta_i$. We let $u_i=p_i$ (for $1 \leq i \leq k$) to be more consistent with some of the papers on this subject.
Let $||\alpha|| = \max_i |\alpha|$ and $||\beta||=\max_i |\beta_i|$. We will assume without loss of generality $||\alpha|| \geq ||\beta||$. We use two different estimates that together will give us a bound on $||\beta||$.
First, we give a bound on $|x|_p$ in terms of $||\alpha||$. In particular, let
$X = (\log |x|_{p_0}, \log |x|_{p_1}, \ldots, \log |x|_{p_k})$, and let
\[{\mathcal L}= ( \log |u_i|_{p_j} )_{1 \leq i \leq k, 0 \leq j \leq k}. \]
Then we have $X = {\mathcal L} \alpha.$ It is easy to see that ${\mathcal L}$ has a left inverse. Let ${\mathcal L^*}$ be such an inverse. Then, we get ${\mathcal L^*} X = \alpha$, and in particular
\[ ||\alpha|| = ||{\mathcal L^*} \cdot X || \leq ||{\mathcal L^*}|| \cdot ||X||. \] Therefore, we get $||X || \geq ||\alpha||/C_H$, and note that $C_H$ is totally independent on $\alpha$ and $\beta$ (although it does depend on few choices we make (such as, which inverse we choose). Hajdu has a paper on how one can optimize the value of $C_H$, as result, we call $C_H$ the Hajdu's constant here, even though I don't use his techniques yet.)
Now, let $p_m$ be such that $\log |x|_{p_m} \leq \log |x|_p$ for all $p \in S$. Since $\sum_p \log |x|_p = 0$, we get that $\log |x|_{p_m} \leq - {||X|| \over k} \leq -{||\alpha||\over C_H k}$. Therefore we get $|x|_{p_m} \leq e^{- ||\alpha|| \over C_H k}$. Notice that $|x|_{p_m} \leq 0.5.$
Now there are two cases to consider, when $p_m=\infty$ and $p_m \neq \infty$. In both cases, we get $|x|_{p_m} \simeq |-\log (1-x)|_{p_m}$ (when $p_m \not = \infty,$ it is equality). Therefore
\[|x|_{p_m} \simeq |\log_{p_m} (1-x)|_{p_m} = |\log y|_{p_m} = |\sum_i \beta_i \log_{p_m} p_i | . \] Using bounds of the linear forms of logarithm from Baker and Yu, we get
$|\sum_i \beta_i \log{p_m} p_i|>||\beta||^{-C_{S,p}}$. Putting these two together, we get
\[ -C_{S,p_m} \log ||\beta|| < \log |x|_{p_m} < -{||\alpha|| \over C_H k}. \]
Multiplying by negative one, we get
\[ ||\alpha||Clearly, $A$ grows much faster than $\log(A)$, therefore the above inequality gives us a bound for $||\alpha||$. In fact, a quick computation tells that if we let $K=kC_H C_{S,p_m}$, then the above inequality forces $||\alpha|| \leq K \log K (1+\epsilon)$.
Applying the above method for the case $S=\{ \infty, 2,3,5,7,11,13,17\}$ and using the Mateev and Yu's results on linear forms of logarithms, we get $||\alpha|| < 2.6 \times 10^{24}$.
In the next posts, I will try to explain how we can lower this bound considerably.
An $S$-unit equation is an equation of the form $ax+by=1$ with $a, b$ given, $x,y \in O_S$ and $x$ and $y$ coprime to each other. It is well known that if $a,b$ and $S$ are given, then we can effectively solve the $S$-unit equation $ax+by=1$, in the sense that we can find an explicit bound using Baker's effective linear forms of logarithms. In fact, with the work of Smart, Wildanger, et. al., it seems possible to solve most reasonable $S$-unit equations (i.e. when $S$ is not too large.)
I will try to put the details of how this can be done (as far as I understand it) in this post. To simplify the exposition, we will work over rationals, and we let $a=b=1$. (I think generalizing to other cases is clear.) In this case, given $S=\{p_0=\infty, p_1,p_2,\ldots,p_k\}$, we are asked to find $\alpha=(\alpha_1,\ldots,\alpha_k)^t$ and $\beta=(\beta_1,\ldots,\beta_k)^t$ so that
\[ \prod_{i=1}^k p_i^{\alpha_i} + \prod_{i=1}^k p_i^{\beta_i} = 1. \]
Therefore $v_i(x)=\alpha_i$ and $v_i(y)=\beta_i$. We let $u_i=p_i$ (for $1 \leq i \leq k$) to be more consistent with some of the papers on this subject.
Let $||\alpha|| = \max_i |\alpha|$ and $||\beta||=\max_i |\beta_i|$. We will assume without loss of generality $||\alpha|| \geq ||\beta||$. We use two different estimates that together will give us a bound on $||\beta||$.
First, we give a bound on $|x|_p$ in terms of $||\alpha||$. In particular, let
$X = (\log |x|_{p_0}, \log |x|_{p_1}, \ldots, \log |x|_{p_k})$, and let
\[{\mathcal L}= ( \log |u_i|_{p_j} )_{1 \leq i \leq k, 0 \leq j \leq k}. \]
Then we have $X = {\mathcal L} \alpha.$ It is easy to see that ${\mathcal L}$ has a left inverse. Let ${\mathcal L^*}$ be such an inverse. Then, we get ${\mathcal L^*} X = \alpha$, and in particular
\[ ||\alpha|| = ||{\mathcal L^*} \cdot X || \leq ||{\mathcal L^*}|| \cdot ||X||. \] Therefore, we get $||X || \geq ||\alpha||/C_H$, and note that $C_H$ is totally independent on $\alpha$ and $\beta$ (although it does depend on few choices we make (such as, which inverse we choose). Hajdu has a paper on how one can optimize the value of $C_H$, as result, we call $C_H$ the Hajdu's constant here, even though I don't use his techniques yet.)
Now, let $p_m$ be such that $\log |x|_{p_m} \leq \log |x|_p$ for all $p \in S$. Since $\sum_p \log |x|_p = 0$, we get that $\log |x|_{p_m} \leq - {||X|| \over k} \leq -{||\alpha||\over C_H k}$. Therefore we get $|x|_{p_m} \leq e^{- ||\alpha|| \over C_H k}$. Notice that $|x|_{p_m} \leq 0.5.$
Now there are two cases to consider, when $p_m=\infty$ and $p_m \neq \infty$. In both cases, we get $|x|_{p_m} \simeq |-\log (1-x)|_{p_m}$ (when $p_m \not = \infty,$ it is equality). Therefore
\[|x|_{p_m} \simeq |\log_{p_m} (1-x)|_{p_m} = |\log y|_{p_m} = |\sum_i \beta_i \log_{p_m} p_i | . \] Using bounds of the linear forms of logarithm from Baker and Yu, we get
$|\sum_i \beta_i \log{p_m} p_i|>||\beta||^{-C_{S,p}}$. Putting these two together, we get
\[ -C_{S,p_m} \log ||\beta|| < \log |x|_{p_m} < -{||\alpha|| \over C_H k}. \]
Multiplying by negative one, we get
\[ ||\alpha||
Applying the above method for the case $S=\{ \infty, 2,3,5,7,11,13,17\}$ and using the Mateev and Yu's results on linear forms of logarithms, we get $||\alpha|| < 2.6 \times 10^{24}$.
In the next posts, I will try to explain how we can lower this bound considerably.
