Let $K$ be a function field over $\mathbb{C}$, i.e. a finitely generated extension of $\mathbb{C}$ of transcendence degree 1. Suppose that $x, y \in K^\ast$ are such that $x + y = 1$. Then the Stother-Mason Theorem states that if $x \not \in \mathbb{C}$ we have $$ H_K(x) \leq |S| + 2g_K - 2, $$ where $g_K$ is the genus of $K$ and $S := \{v \in M_K : v(x) \neq 0 \text{ or } v(y) \neq 0\}$ are the places of $K$ such that $v(x) \neq 0$ or $v(y) \neq 0$.

The Stother-Mason Theorem can be used to count the number of solutions to unit equations in function fields, but it leads to the appearance of $g$ in the resulting upper bounds. Therefore I am wondering if the upper bound in the Stother-Mason Theorem can be replaced by $$ H_K(x) \leq c \gamma |S|, $$ where $c$ is some absolute constant and $\gamma$ is the gonality, that is $$ \gamma := \min_{t \in K} [K : \mathbb{C}(t)]. $$ Note that this upper bound is weaker than the Stother-Mason Theorem if $|S|$ is of size at least $2g_K - 2$, but a lot stronger if $|S|$ is very small.

So far I have been able to find the following: Brownawell and Masser in their paper ``vanishing sums in function fields'' find examples for every value of $g$ such that equality holds in the Stother-Mason Theorem for infinitely many values of $|S|$. Unfortunately $|S|$ is of size at least $g$ in their examples, so it does not say anything about the truth of $H_K(x) \leq c \gamma |S|$.

Edit: changed Mason's Theorem to Stother-Mason Theorem.

Let $K$ be a function field over $\mathbb{C}$, i.e. a finitely generated extension of $\mathbb{C}$ of transcendence degree 1. Suppose that $x, y \in K^\ast$ are such that $x + y = 1$. Then the Stothers-Mason Theorem states that if $x \not \in \mathbb{C}$ we have $$ H_K(x) \leq |S| + 2g_K - 2, $$ where $g_K$ is the genus of $K$ and $S := \{v \in M_K : v(x) \neq 0 \text{ or } v(y) \neq 0\}$ are the places of $K$ such that $v(x) \neq 0$ or $v(y) \neq 0$.

The Stothers-Mason Theorem can be used to count the number of solutions to unit equations in function fields, but it leads to the appearance of $g$ in the resulting upper bounds. Therefore I am wondering if the upper bound in the Stothers-Mason Theorem can be replaced by $$ H_K(x) \leq c \gamma |S|, $$ where $c$ is some absolute constant and $\gamma$ is the gonality, that is $$ \gamma := \min_{t \in K} [K : \mathbb{C}(t)]. $$ Note that this upper bound is weaker than the Stothers-Mason Theorem if $|S|$ is of size at least $2g_K - 2$, but a lot stronger if $|S|$ is very small.

So far I have been able to find the following: Brownawell and Masser in their paper ``vanishing sums in function fields'' find examples for every value of $g$ such that equality holds in the Stothers-Mason Theorem for infinitely many values of $|S|$. Unfortunately $|S|$ is of size at least $g$ in their examples, so it does not say anything about the truth of $H_K(x) \leq c \gamma |S|$.

Edit: changed Mason's Theorem to Stothers-Mason Theorem.

Let $K$ be a function field over $\mathbb{C}$, i.e. a finitely generated extension of $\mathbb{C}$ of transcendence degree 1. Suppose that $x, y \in K^\ast$ are such that $x + y = 1$. Then Mason's Theorem states that if $x \not \in \mathbb{C}$ we have $$ H_K(x) \leq |S| + 2g_K - 2, $$ where $g_K$ is the genus of $K$ and $S := \{v \in M_K : v(x) \neq 0 \text{ or } v(y) \neq 0\}$ are the places of $K$ such that $v(x) \neq 0$ or $v(y) \neq 0$.

Mason's Theorem can be used to count the number of solutions to unit equations in function fields, but it leads to the appearance of $g$ in the resulting upper bounds. Therefore I am wondering if the upper bound in Mason's Theorem can be replaced by $$ H_K(x) \leq c \gamma |S|, $$ where $c$ is some absolute constant and $\gamma$ is the gonality, that is $$ \gamma := \min_{t \in K} [K : \mathbb{C}(t)]. $$ Note that this upper bound is weaker than Mason's Theorem if $|S|$ is of size at least $2g_K - 2$, but a lot stronger if $|S|$ is very small.

So far I have been able to find the following: Brownawell and Masser in their paper ``vanishing sums in function fields'' find examples for every value of $g$ such that equality holds in Mason's Theorem for infinitely many values of $|S|$. Unfortunately $|S|$ is of size at least $g$ in their examples, so it does not say anything about the truth of $H_K(x) \leq c \gamma |S|$.

Let $K$ be a function field over $\mathbb{C}$, i.e. a finitely generated extension of $\mathbb{C}$ of transcendence degree 1. Suppose that $x, y \in K^\ast$ are such that $x + y = 1$. Then the Stother-Mason Theorem states that if $x \not \in \mathbb{C}$ we have $$ H_K(x) \leq |S| + 2g_K - 2, $$ where $g_K$ is the genus of $K$ and $S := \{v \in M_K : v(x) \neq 0 \text{ or } v(y) \neq 0\}$ are the places of $K$ such that $v(x) \neq 0$ or $v(y) \neq 0$.

The Stother-Mason Theorem can be used to count the number of solutions to unit equations in function fields, but it leads to the appearance of $g$ in the resulting upper bounds. Therefore I am wondering if the upper bound in the Stother-Mason Theorem can be replaced by $$ H_K(x) \leq c \gamma |S|, $$ where $c$ is some absolute constant and $\gamma$ is the gonality, that is $$ \gamma := \min_{t \in K} [K : \mathbb{C}(t)]. $$ Note that this upper bound is weaker than the Stother-Mason Theorem if $|S|$ is of size at least $2g_K - 2$, but a lot stronger if $|S|$ is very small.

So far I have been able to find the following: Brownawell and Masser in their paper ``vanishing sums in function fields'' find examples for every value of $g$ such that equality holds in the Stother-Mason Theorem for infinitely many values of $|S|$. Unfortunately $|S|$ is of size at least $g$ in their examples, so it does not say anything about the truth of $H_K(x) \leq c \gamma |S|$.

Edit: changed Mason's Theorem to Stother-Mason Theorem.