Koizumi–Liu Eventual Sign Alternation Conjecture

Koizumi constructs a representable simple rank-$6$ matroid $M$ for which $$\operatorname{Mag}(M;q)$$ has a pole of order $4$ at $q=-1$ but a pole of order $5$ at $q=i$. After writing $$A_M(t)=\operatorname{Mag}(M;-t),$$ the higher-order pole at $t=i$ forces the Taylor coefficients of $A_M(t)$ to fail eventual nonnegativity. Consequently, $$(-1)^\ell[q^\ell]\operatorname{Mag}(M;q)<0$$ for infinitely many $\ell$. The matroid is representable by twelve vectors in $\mathbb R^6$, so it yields an actual real central hyperplane arrangement and therefore directly disproves Koizumi–Liu's conjecture for real arrangements.

Koizumi constructs a representable simple rank-$6$ matroid $M$ for which $$\operatorname{Mag}(M;q)$$ has a pole of order $4$ at $q=-1$ but a pole of order $5$ at $q=i$. After writing $$A_M(t)=\operatorname{Mag}(M;-t),$$ the higher-order pole at $t=i$ forces the Taylor coefficients of $A_M(t)$ to fail eventual nonnegativity. Consequently, $$(-1)^\ell[q^\ell]\operatorname{Mag}(M;q)<0$$ for infinitely many $\ell$. The matroid is representable by twelve vectors in $\mathbb R^6$, so it yields an actual real central hyperplane arrangement and therefore directly disproves Koizumi–Liu's conjecture for real arrangements.