Around Kouchnirenko's conjecture I

Definitions and mathematical statement

Fix positive integers $m_1,m_2$. Consider two real polynomials $f_i\in\mathbb R[x,y]$, each with at most $m_i$ nonzero monomial terms. The exponents are nonnegative integers, with no degree bound. A common zero $(u,v)$ is simple, or nondegenerate, when \[ f_1(u,v)=f_2(u,v)=0,\qquad \det\begin{pmatrix}(f_1)_x&(f_1)_y\$f_2)_x&(f_2)_y\end{pmatrix}(u,v)\ne0. \] Let $N_+(m_1,m_2)$ be the supremum of the number of such zeros in the strictly positive quadrant $(0,\infty)^2$ over all allowed polynomial pairs. The problem asks for a useful, ideally sharp, upper bound for $N_+(m_1,m_2)$ depending only on the term counts, and examples showing how close the bound is to optimal.

Zeros on either coordinate axis are excluded. Multiplicities are not counted, and positive-dimensional common-zero sets do not contribute nondegenerate zeros. A common monomial factor may be removed without changing the positive solutions, illustrating why total degree is not the relevant measure of complexity.

The naive product $(m_1-1)(m_2-1)$ is not a valid general upper bound: the source discusses counterexamples and the sharp five-solution result for two trinomials. General fewnomial estimates give finite bounds, but the question seeks stronger estimates reflecting the separate term counts. A Bézout estimate that grows with unrestricted degrees does not provide the requested uniform control.

Short English statement

Find sharp or substantially improved bounds for positive simple zeros of two polynomials using only their numbers of terms.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 4700028 (AMR-046-0028), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Gasull - Some open problems in low dimensional dynamical systems (2020), Problem environment 28: Around Kouchnirenko's conjecture I. Original source indexed by AMR-046-0028. https://arxiv.org/abs/2012.02524.