Kouchnirenko's fewnomial conjecture for positive solutions

Consider a system of $n$ polynomials in $n$ variables, where the $i$ th polynomial has $m_i$ monomials and all solutions are simple. Kouchnirenko's conjecture. The system has at most

(m_1-1)(m_2-1)\cdots(m_n-1)

solutions with positive coordinates. This conjecture seeks a substantially sharper bound for positive solutions than general fewnomial estimates; the source notes that its attribution has passed into folklore and does not establish whether it is resolved.

Sources & referencesView supporting material

Primary source

Frank Sottile, “Enumerative Real Algebraic Geometry”, arXiv:math/0107179 (2002).

fewnomials real-algebraic-geometry

1 week ago

Progress summary

Refreshed 7 hours ago

Solved

A documented counterexample shows that the proposed bound is false, although a broader question about polynomial bounds for positive solutions remains open.

Kouchnirenko posed the conjecture in the late 1970s: a system whose $i$ th polynomial has $m_i$ monomials should have at most $\prod_i(m_i-1)$ nondegenerate positive roots. The exact conjecture is false.

Known results

Haas (2000) exhibited a pair of bivariate trinomials with at least $5$ positive roots.
Li, Rojas, and Wang (2003) proved that bivariate trinomial systems have at most $5$ isolated positive roots.

2006 counterexample

The paper Extremal Real Algebraic Geometry and $\mathcal{A}$ -Discriminants gives a simpler bivariate system of two trinomials with exactly $5$ nondegenerate positive roots, exceeding the conjectured bound $(3-1)(3-1)=4$ . Its numerical verification uses Smale’s Alpha Theory. A related broader polynomial-in-support-size bound remains open, even in the bivariate case.

Current status (as of August 2026): The exact conjecture is resolved negatively by the verified $5$ -root counterexample, while broader fewnomial-bound questions remain open.

Sources

No solutions have been posted yet.