Every equality between convergent periods follows from the standard formal relations for periods: linearity of integration, additivity with respect to domains, and algebraic changes of variables. More precisely, if P=int_Df(x_1,…,x_n),dx_1… dx_n and Q=int_Eg(y_1,…,y_m),dy_1… dy_m are convergent integrals of rational functions with coefficients in Q over domains D⊆R^n and E⊆R^m defined by polynomial inequalities with coefficients in Q, and if P=Q as complex numbers, then this equality is derivable from those formal relations.

Kontsevich–Zagier conjecture

Every equality between convergent periods follows from the standard formal relations for periods: linearity of integration, additivity with respect to domains, and algebraic changes of variables. More precisely, if $P=\int_{D}f(x_1,\ldots,x_n)\,dx_1\cdots dx_n$ and $Q=\int_{E}g(y_1,\ldots,y_m)\,dy_1\cdots dy_m$ are convergent integrals of rational functions with coefficients in $\mathbb{Q}$ over domains $D\subseteq\mathbb{R}^n$ and $E\subseteq\mathbb{R}^m$ defined by polynomial inequalities with coefficients in $\mathbb{Q}$ , and if $P=Q$ as complex numbers, then this equality is derivable from those formal relations.

References

Primary source

arXiv

Additional references

Kontsevich-Zagier conjecture for Hensel-minimal fields with sections — arXiv

Posted 4 weeks ago

Progress summary

Refreshed 4 weeks ago

Claimed progress

A new technical framework strengthens work around the conjecture, but no source reports a proof or disproof.

The Kontsevich–Zagier conjecture predicts that equalities between periods follow from the standard formal rules for periods. The general conjecture remains unsettled; the latest work extends machinery relevant to it without claiming a complete proof.

Known results

Huber and Wüstholz proved the conjecture for classical periods of $1$ -motives over $\mathbb{Q}$ ; a 2025 preprint also develops related $p$ -adic results at depths $1$ and $2$ .

September 2026 framework

On September 8, 2026, the article Kontsevich-Zagier conjecture for Hensel-minimal fields with sections reported a new framework for motivic integration over Hensel-minimal fields and proved compatibility with the Cluckers–Loeser integral. This is claimed progress on supporting machinery, not a resolution of the conjecture.

Current status (as of September 2026): The conjecture is open; substantial special-case and framework results are known, but no general proof or counterexample was found.

Sources

No solutions have been posted yet.