Bipartite perfect matching as a real polynomial

Gal Beniamini*, Noam Nisan

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


We obtain a description of the Bipartite Perfect Matching decision problem as a multilinear polynomial over the Reals. We show that it has full total degree and (1−o(1))⋅2n2 monomials with non-zero coefficients. In contrast, we show that in the dual representation (switching the roles of 0 and 1) the number of monomials is only exponential in Θ(n log n). Our proof relies heavily on the fact that the lattice of graphs which are “matching-covered” is Eulerian.

Original languageAmerican English
Pages (from-to)91-131
Number of pages41
JournalIsrael Journal of Mathematics
Issue number1
StatePublished - Sep 2023

Bibliographical note

Funding Information:
We thank Nati Linial for helpful discussions. This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 740282).

Publisher Copyright:
© 2023, The Hebrew University of Jerusalem.


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