# Solving not so much overdetermined system of multivariate polynomial equations

I'm studying algorithms solving multivariate equations. I'm stuck in solving overdetermined set of quadratic equations. Concretely, with the number $$n$$ of variables, the number of equations is $$m=\epsilon n^2$$. If $$\epsilon\geq 1/2$$, it is known that trivial linearization solves the set of equations. However, in the case that $$\epsilon<1/2$$, the trivial linearization seems not working anymore.

But there are some papers suggesting that if $$\epsilon$$ is somewhat large ($$1/16<\epsilon<1/2$$) then the system of equations can be solved in $$O(n^{2\omega})$$-time where $$2\leq\omega\leq3$$ is linear algebra constant. Efficient algorithms for solving overdefined systems of multivariate polynomial equations claimed that the time complexity of XL algorithm is $$O(n^{\omega\lceil\frac 1 {\sqrt{\epsilon}}\rceil})$$. It implies that if $$1/4\leq \epsilon<1/2$$ then the time complexity is $$O(n^{2\omega})$$. But I cannot catch the case.

One more paper here. Complexity of Groebner basis computation for semi-regular overdetermined sequences over F_2 with solutions in F_2 claimed that the degree of regularity is asymptotically $$\max(1/8\epsilon,2)$$ which implies that if $$1/16\leq \epsilon<1/2$$ then the time complexity of F5 algorithm is $$O(n^{2\omega})$$.

For both cases, I don't know how the algorithms can get such time complexity (when maximal degree of XL or F5 computation is 2). Are those my missings or just asymptotics not for the cases?