MathDB

Matkowski’s conjecture

Let K\mathbb{K} be the coefficient field. Matkowski’s conjecture asserts that every two-variable formal power series M∈K[[x,y]]M\in\mathbb{K}[[x,y]] satisfying the weak associativity equation M(M(x,y),x)=M(x,M(y,x))M(M(x,y),x)=M(x,M(y,x)) and having a non-symmetric linear part, meaning M1(x,y)≠M1(y,x)M_1(x,y)\ne M_1(y,x), is linear, i.e. has no terms of total degree greater than one.

References

Primary source

Aequationes mathematicae

Additional references

algebraformal-power-seriesfunctional-equations
Posted

Progress summary

Refreshed
Claimed solved

A 2026 preprint claims the conjecture is false by giving nonlinear continuous solutions, while a new journal paper’s result is not yet known.

Matkowski’s conjecture concerns continuous solutions of f(f(−x)+x)=f(−f(x))+f(x)f(f(-x)+x)=f(-f(x))+f(x) and predicts that every solution is linear on each half-line. It arose from Janusz Matkowski’s work on translative, weakly associative operations.

February–September 2026 developments

  • On February 17, 2026, Tibor Kiss’s preprint A Counterexample to Matkowski’s Conjecture for Quasi Graph-Additive Functions claimed continuous solutions with nonlinear components, directly refuting the conjecture.
  • On April 29, 2026, Kiss’s follow-up preprint claimed broader solution families and characterizations, rather than restoring the conjecture.
  • On September 4, 2026, Amr Zakaria’s article On Matkowski’s conjecture for weakly associative formal power series was published online, but the retrieved metadata gives no theorem statement, so its effect is unknown.

Current status (as of September 2026): A counterexample is claimed for the continuous functional-equation formulation but remains unverified, and the September journal article cannot yet be assessed.

Sources

Solutions 0

No solutions have been posted yet.