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 magnituge 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 $\alpha_1, \ldots, \alpha_n$.

UPD1: it seem that this question is rather difficult. Let us consider a particular case: when all $\alpha_i$ are quadratic irrationality. Is it possible to prove the required result about the measure of its linear independence?

UPD2: Is it possible to prove the statement under some reasonable hypothesis like effective abc-conjecture or Vojta-conjecture?

UPD3: It is possible to prove that there exist some $C$ such that $|\alpha_1 + \alpha_2| \ge H^{-C(\deg(p_1) + \deg(p_2))}$

(if $\alpha_1 + \alpha_2 \not=0$)?


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".

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  • $\begingroup$ As I understand your estimate gives something like $H^{-\deg(\alpha_1)\cdot\dots\cdot\deg(\alpha_n)}$. $\endgroup$ – Alexey Milovanov May 20 '19 at 13:17
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    $\begingroup$ However, I wanted estimate $H^{q (\deg(\alpha_1) + \ldots + \deg(\alpha_n))}$, where $q$ is a fixed polynomial,i.e. $q$ does not depend on $n$ and $\deg (\alpha_i)$. $\endgroup$ – Alexey Milovanov May 20 '19 at 13:20
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    $\begingroup$ If $q$ can not depend on $n$, then you can find a counterexample already with rational numbers. $\endgroup$ – Oleg Eroshkin May 20 '19 at 13:28
  • $\begingroup$ So, for every polynomial $p$ and for every $n$ there are $n$ rational numbers $\alpha_1, \ldots, \alpha_n$ such that $|\alpha_1 + \ldots \alpha_n| \ge H^{- p(n)}$, where $H$ is the maximum value among numerators and denominators of $\alpha_i$? $\endgroup$ – Alexey Milovanov May 20 '19 at 13:39
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    $\begingroup$ Yes, you are right. For a minute I was sure that deg of a rational number is 0. $\endgroup$ – Oleg Eroshkin May 20 '19 at 13:55

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