I would like to find effective upper bound for the height of a-b in terms of the heights of a and b. Thanks in advance. Here the height of an algebraic number a is the maximum of the absolute values of the coefficients of the minimal polynomial of a over Z (and of course, primitive).
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2$\begingroup$ This is not possible unless you bound the degrees of a and b as well. The trace of $a+1$ is $\text{tr}(a)+\text{deg}(a)$. $\endgroup$– Michael ZieveCommented Jun 29, 2014 at 5:49
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1$\begingroup$ Related question: mathoverflow.net/questions/64643/… $\endgroup$– Michael ZieveCommented Jun 29, 2014 at 5:59
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$\begingroup$ You can get such an estimate for the $d$'th root of the largest coefficient, where $d$ is the degree of the minimal poly. There are standard estimates relating the usual height $H_1(a)$ (not your quantity) and $H_2(a)$ the $d$'th root your height. (See the answers to the question that Michael Zieve gives the link.) For $H_1$, one has (if I remember correctly) $H_1(a-b)\le2H_1(a)H_1(b)$. From this one can get a similar estimate for $H_2$. $\endgroup$– Joe SilvermanCommented Jun 29, 2014 at 12:05
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$\begingroup$ Unfortunately, I have no Waldschmidt book with the relations between H(a) and h(a) (weyl height). With the estimates given in Zieve link I got H(a-b)<< H(a)^H(b) or something like that. $\endgroup$– Diego MarquesCommented Jun 29, 2014 at 19:33
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