3 problems

Let a straight-line program for a polynomial in one variable be a list … where each $s_i$, for $i\geq 2$, is one of $s_j+s_k$, $s_j-s_k$, or $s_js_k$ for some $j,k<i$. For…

Congruence-restricted $\tau$-hypothesis. There is an absolute constant $c_p>0$ such that, for every $f\in\mathbb{Z}[x_1]\setminus\{0\}$, the number of such roots is at most

Shub–Smale $\tau$-conjecture. There is an absolute constant $\kappa\geq 1$ such that, for every $f\in\mathbb{Z}[x_1]\setminus\{0\}$, the number of distinct roots of $f$ in…