Estimation of a sum of algebraic numbers Let $\alpha_1, \ldots, \alpha_n$ be algebraic numbers and let $p_1, \ldots, p_n$ be the corresponding minimal polynomials with integer coefficients. Denote by $H$ the maximal magnitude among all coefficients of all $p_i$.
Assume that $\alpha_1 + \ldots + \alpha_n \not= 0$. Is it true that
$$|\alpha_1 + \ldots + \alpha_n | \ge {(CH)}^{-q(\text{deg}(p_1) +\ldots \text{deg}(p_n) )}$$ for some fixed polynomial $q$ and constant $C$?
Warning: $C$ and $q$ should not depend on $\alpha_1, \ldots, \alpha_n$.
Updates, since the question appears difficult:

*

*Does the statement hold when all $\alpha_i$ are quadratic irrationals?

*Does the statement hold under reasonable hypotheses like the effective $abc$ conjecture or Vojta conjecture?

*Is there a $C$ such that $|\alpha_1 + \alpha_2| \ge H^{-C(\deg(p_1) + \deg(p_2))}$ whenever $\alpha_1 + \alpha_2 \not=0$?

 A: In the answer below I assume that $q$ may depend on $n$. It seems plausible that the assertion is true for $q(x)=x^2$ or something similar, but I don't know how to prove it. It may be hard. The answer below is based on Liouville's inequality, that is on estimates for $P(\alpha_1,\dots,\alpha_n)$
which are essentially the best possible for arbitrary polynomials and algebraic numbers. But for a fixed $P$ much better estimates can be available as illustrated by Roth's theorem. Unfortunately, Roth's theorem and Schmidt's Subspace theorem are ineffective, so can't be used here.
Old answer:
Yes, that's true. I will write the estimates in terms of height.
The usual height $H(p)$ of a polynomial $p$ with integer coefficients is defined as
the maximum of absolute values of its coefficients.
Then the usual height $H(\alpha)$ of an algebraic number $\alpha$ is the usual height of its minimal polynomial. The absolute logarithmic height of $\alpha$ is
$$h(\alpha)=\frac{1}{d}\log H(\alpha)\,,$$
where $d$ is the degree of $\alpha$.
The basic version of Liouville's inequality says that if $\alpha\neq0$ then
$$|\alpha|\geq\frac{1}{H(\alpha)}\,.$$
Now, combine that with
$$h(\alpha_1+\dots+\alpha_n)\leq\log n+h(\alpha_1)+\dots+h(\alpha_n)$$
and $$\deg(\alpha_1+\dots+\alpha_n)\leq \deg(\alpha_1)\cdot\dots\cdot\deg(\alpha_n)\,.$$
All these claims about height can be found in many places. I like Chapter 3 of Waldschmidt "Diophantine approximation on linear algebraic groups".
