I would like to find effective upper bound for the height of $a+b$ and $a/b$ and $ab$ knowing the heights of $a$ and $b$. Thanks.

If you know only heights of $a$ and $b$, you may estimate heights of $a+b$, $a/b$ and $ab$. Assuming that $h$ is an absolute (Weil) height: $$h(ab)\leq h(a)+h(b)$$ $$h(a/b)\leq h(a)+h(b)$$ $$h(a+b)\leq\log 2 +h(a)+h(b)$$ This bounds are sharp. You may find this, for example, in M. Waldschmidt "Diophantine approximation on linear algebraic groups", Chapter 3. 


There are standard estimates for the heights of algebraic numbers $a_1,...,a_n$ in terms of their elementary symmetric functions $s_1,...,s_n$, or equivalently, estimates relating the heights of the roots of a polynomial to the heights of its coefficients. You can find this in many textbooks, including for example my Arithmetic of Elliptic Curves (Theorem VIII.5.9) or Lang's Fundamentals of Diophantine Geometry (Chapter 3, Section 2). The estimate is $$ \sum_{i=1}^n h(a_i)  n\log(2) \le h([1,s_1,...,s_n]) \le \sum_{i=1}^n h(a_i) + (n1)\log(2). $$ 


If $a$ is the root of a characteristic polynomial of $M$ and $b$ is the root of the characteristic polynomial of $N,$ then $ab$ is a root of the charcteristic polynomial of $M \otimes N,$ and $a+b$ is a root of the characteristic polynomial of $M \otimes I + I \otimes N.$ That should be enough to compute the height. 


I too was looking for the answer to the same question. It seems necessary to define "height" since there are of course several variants in use. The height $h(a)$ I am interested in is the maximum of the absolute values of the coefficients of the minimal polynomial of the algebraic number $a$. Note that the first answer refers to a different notion of height, since, for example, $9 = h(9) = h(3 \cdot 3) \not \leq h(3) + h(3) = 3 + 3 = 6.$ I imagine there must be upper bounds of the form $h(ab) \leq f(d) h(a)^{g(d)} h(b)^{g(d)}$ for some simple functions $f$ and $g$, where $d$ is the degree of a field extension of $\mathbb{Q}$ containing both $a$ and $b$, for example. Are there any such results in the literature, and similarly for $h(a+b)$ and $h(a/b)$? 

