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Are there general methods (in number theory, algebraic geometry?) to find integral solutions of $x^p+y^q+z^r+cdots=t^s$?

I’ve already seen methods of solving equations of the form $x^p+y^q=z^r$ using the fact that $$2^a3^b+2^{a+1}3^b=2^a3^{b+1}.$$

Similarly, can we let, for $A_iin mathbb{Z}$, $$Omega =sum_{j_1,dots,j_k}a_{j_1,…,j_k}prod_{1leq ileq k} A_i^{j_i},$$ and consider $$sum_i B^{p_i}_i=beta^q$$ (for $beta,B,p,q$ in the same field) comparing this to $$Omega =prod A_i^{k_i}?$$ I’m not sure I stated this correctly or fully: for example, could we compare

$$eta^a(1+eta+Aeta^2)^b+eta^{a+1}(1+eta+Aeta^2)^b+Aeta^{a+2}(1+eta+Aeta^2)^b=eta^a(1+eta+Aeta^2)^{b+1}$$ to $$x^p+y^q+Az^r=t^s?$$ Over finite fields? Are there simpler methods?

Yours, Alex

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nt.number theory – P-adic distance between solutions to S-unit equation

Let $p$ be a fixed prime number and $S$ is a finite set of prime numbers which does not contain $p$. A theorem of Siegel asserts that the number of solutions to the $S$-unit equations are finite; that is, there are only finitely many $S$-unit $u$ such that $1-u$ is also an $S$-unit. Therefor for each such $S$ there exist a lower bound on $|u_1-u_2|_p$ where $u_1$ and $u_2$ are solutions to $S$-unit equations.

My question is: does there exist such a lower bound uniformly? More precisely, does there exist a lower bound for the $p$-adic distance between solutions to the $S$-unit equations that only depends on the size of $S$(and perhaps on $p$)? Here we are assuming $S$ does not contain $p$.

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algebra precalculus – Solutions of third order equation

I’m looking for which values of $rho >0$, the following holds
begin{align}
– left( (b_2 + a kappa) + (1+kappa) (b_1 + a kappa) right) rho^2 + mu rho – c kappa geq 0 ~(1).
end{align}

where $a$, $b_1$, $b_2$, $c$ and $mu$ are positive constants and $kappa > 0$ is a parameter that I can freely set to any positive value.

The discriminant of the quadratic equation is
begin{align}
Delta = mu^2 – 4 c kappa left( (b_2 + a kappa) + (1+kappa) (b_1 + a kappa) right) ~(2).
end{align}

To ensure that I can find $rho > 0$ such that Eq. (1) is satisfied, I need to impose that $Delta > 0$. Thus, I end up looking for which values of $kappa > 0$, Eq. (2) holds. However, Eq. (2) is a third order equation and not sure if analytically I can find the set of values of $kappa > 0$ to satisfy Eq. (2).

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