Algebraična reformulacija Bessis-Moussa-Villanijeve domneve je ekvivalentna družini neenakosti za sledi, ki veljajo za vse dimenzije in vsebujejo pozitivno semidefinitne matrike. Zadostne pogoje, da ...to velja v obliki algebraičnih identitet, ki vsebujejo polinome v nekomutativnih spremenljivkah, sta podala Klep in Schweighofer ["Sums of Hermitian squares and the BMV conjecture", J. Stat. Phys. 133, 739 (2008)]. Kasneje je bil obstojteh certifikatov dokazan ali ovržen za vse primere razen enega, ki pa je rešen v tem delu.
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