= Schinzel-Tijdeman exponent theorem
{c}
{title2=$y^m=f(x),\quad |y|>1\Longrightarrow m\le C(f)$}
For a fixed <polynomial> $f\in\mathbb Q[X]$ with at least two distinct complex roots, <integer> solutions with $m\ge2$ and $|y|>1$ have an effectively bounded exponent. Factorization over a <splitting field> and the fixed differences between distinct roots control common <ideal> <divisors>. Outside a fixed set of primes, each factor's valuation is divisible by $m/\gcd(m,e_i)$, where $e_i$ is its root <multiplicity>. The generalized power-equation estimates from <linear forms in logarithms of algebraic numbers>, including <unit> and exceptional-prime factors, give the exponent bound. A single-root <polynomial> and the values $y=0,\pm1$ cannot be included: $4^m=(2^m)^2$, and $1^m=f(0)$ for $f(X)=X(X-1)+1$, give infinite exponent families.
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