Exactness of the upper bound on singular Newton homotopy curves

Determine whether, for every pair of integers d_1 \geq d_2 with d_1 \geq 2, there exists a bivariate polynomial system satisfying the paper’s genericity assumptions for which the upper bound on the number of singular Newton homotopy points is attained: \#S=3(d_1-1)^2 when d_1=d_2 and \#S=(d_1+d_2-1)^2-d_1d_2 when d_1>d_2.

Background

For a bivariate polynomial system f=(f_1,f_2) with degrees d_1\geq d_2 and d_1\geq2, the set S consists of start points r for which the associated Newton homotopy curve C_r is singular and r is not a real solution of f. The paper derives degree-dependent upper bounds for #S: 3(d_1-1)2 when the degrees are equal, and (d_1+d_2-1)2-d_1d_2 when d_1>d_2.

The authors exhibit four examples with degree pairs (2,1), (2,2), (3,1), and (3,2) in which the corresponding bounds are exact. They leave unresolved whether every admissible degree pair admits an example attaining the relevant bound.

References

We leave it as an open problem to check whether for all values of $d_1\geq d_2$ with $d_1\geq 2$, there exists an example where the bounds are exact.

— Geometry of Newton homotopies: bivariate case  (2609.30189 - Buettner et al., 24 Sep 2026) in Section 6.1, immediately following Theorem 6.1 (Upper bound on number of singular points)

This suggests the following. Almost all $f$'s are infinity complete.

— Geometry of Newton homotopies: bivariate case  (2609.30189 - Buettner et al., 24 Sep 2026) in Section 6.3, subsection “Upper bound on number of cells when degrees are different,” immediately after Definition 6.3