Questions

$(1)$

Let $f = x_1 x_2 + x_2^2 + x_1^2$ . Compute $u_f$ for ${\alpha}_2 = 2$ , and then for ${\alpha}_2 = 3$ .

$(2)$

More generally, how to choose ${\alpha}_2$ such that $f$ is the only pre-image of $u_f$ ?

$(3)$

Consider now two bivariate polynomials $f$ and $g$ with respective $x_1$ -degrees $d_1$ and $c_1$ . How to choose ${\alpha}_2$ such that $fg$ is the only pre-image of $u_{fg}$ ?

$(4)$

Describe an algorithm reducing the computation of the product $fg$ to a multiplication in ${\mbox{${\mathbb{K}}$}}[x_1]$ . Give an upper bound for its running time in terms of number of operations in ${\mathbb{K}}$ .