Delta-matroid invariance conjecture for ML degrees

We prove Conjecture 1.8 of Janike Oldekop, Euler Stratifications of Second Hypersimplices via Delta-matroids (arXiv:2606.16482), in every dimension. In fact, we obtain an explicit formula for the maximum-likelihood degree depending only on the ranks encoded by the associated delta-matroid.

Let $d\geq3$, let $E=[d]$, and let $C=(c_{ij})_{i,j\in E}$ be a scaling matrix: $C$ is symmetric, $c_{ii}=0$, and $c_{ij}\neq0$ whenever $i\neq j$. Write

$$f_C(x)=\sum_{1\leq i<j\leq d}c_{ij}x_ix_j, \qquad T=\{[x_1:\ldots:x_d]\in\mathbb P^{d-1}:x_i\neq0\text{ for every }i\}. \tag{1}$$

The source's Proposition 2.1 identifies the maximum-likelihood degree with the signed Euler characteristic

$$\operatorname{MLdeg}(V_d^c) =(-1)^d\chi\bigl(\{f_C=0\}\cap T\bigr). \tag{2}$$

For a nonempty subset $S\subseteq E$, put

$$s=|S|, \qquad r(S)=\operatorname{rank}C[S], \qquad Q_S=\{[x_i]_{i\in S}\in\mathbb P^{s-1}:x^{\mathsf T}C[S]x=0\}. \tag{3}$$

We first compute the Euler characteristic of every such coordinate section. A rank-zero quadric is all of $\mathbb P^{s-1}$, hence has Euler characteristic $s$. If $r\geq1$, a linear change of coordinates identifies its projective quadric with the cone over a smooth rank-$r$ quadric, with vertex $\mathbb P^{s-r-1}$. Removing the vertex gives an affine-space bundle over the smooth quadric. If $q_r$ denotes the Euler characteristic of that smooth quadric, then

$$\chi(Q_S)=s-r+q_r. \tag{4}$$

Here $q_1=0$ and $q_2=2$. For $r\geq3$, write the smooth quadric in hyperbolic coordinates as

$$x_1x_2+x_3^2+\cdots+x_r^2=0. \tag{5}$$

The chart $x_1\neq0$ is $\mathbb A^{r-2}$, while the section $x_1=0$ consists of one vertex together with an affine-line bundle over the smooth rank-$(r-2)$ quadric. Therefore

$$q_r=2+q_{r-2} =r-1+\mathbf1_{\{r\text{ even}\}}. \tag{6}$$

Combining (4) and (6), including the rank-zero case separately, gives the uniform formula

$$\chi(Q_S) =|S|-1+\mathbf1_{\{r(S)\text{ even}\}}. \tag{7}$$

Apply inclusion-exclusion to the coordinate hyperplanes in $\mathbb P^{d-1}$. The intersection associated with a surviving nonempty coordinate set $S$ is precisely $Q_S$. Thus

$$\chi\bigl(\{f_C=0\}\cap T\bigr) =\sum_{\emptyset\neq S\subseteq E} (-1)^{d-|S|} \left(|S|-1+\mathbf1_{\{r(S)\text{ even}\}}\right). \tag{8}$$

Since $d\geq2$,

$$\sum_{\emptyset\neq S\subseteq E}(-1)^{|S|}(|S|-1)=1. \tag{9}$$

The missing empty set has rank zero and therefore contributes exactly the same $1$. Equations (2), (8), and (9) consequently yield the explicit all-dimensional formula

$$\boxed{\displaystyle \operatorname{MLdeg}(V_d^c) =\sum_{\substack{S\subseteq E\r(S)\text{ even}}}(-1)^{|S|}.} \tag{10}$$

Equivalently, writing $\nu(C[S])=|S|-r(S)$ and using $\sum_{S\subseteq E}(-1)^{|S|}=0$, we obtain

$$\boxed{\displaystyle \operatorname{MLdeg}(V_d^c) =\frac12\sum_{S\subseteq E}(-1)^{\nu(C[S])}.} \tag{11}$$

It remains to show that every rank in (10) and (11) is determined by the delta-matroid. Let

$$\mathcal F(C) =\{F\subseteq E:\det C[F]\neq0\}, \qquad \det C[\emptyset]=1. \tag{12}$$

For every symmetric matrix $A$, there exists a nonsingular principal submatrix of order $\operatorname{rank}A$. Indeed, if a diagonal entry is nonzero, pivot on its one-element principal block and apply induction to the symmetric Schur complement. If every diagonal entry vanishes but $A\neq0$, choose a nonzero off-diagonal entry $a_{ij}$ and pivot on the nonsingular principal block

$$A[\{i,j\}] =\begin{pmatrix}0&a_{ij}\\a_{ij}&0\end{pmatrix}, \qquad \det A[\{i,j\}]=-a_{ij}^2\neq0. \tag{13}$$

Again the Schur complement is symmetric, and induction produces a nonsingular principal submatrix of the full rank. Applying this to $A=C[S]$ proves

$$r(S) =\max\{|F|:F\in\mathcal F(C),\ F\subseteq S\}. \tag{14}$$

Therefore an isomorphism of delta-matroids carries each subset $S$ to a subset of the same cardinality and the same rank (14). Every summand in (10), equivalently (11), is preserved. Hence

$$\mathcal F(C_1)\cong\mathcal F(C_2) \quad\Longrightarrow\quad \operatorname{MLdeg}(V_d^{c_1}) =\operatorname{MLdeg}(V_d^{c_2}), \tag{15}$$

which proves Conjecture 1.8 for every $d\geq3$.

The degenerate boundary $d=2$ is immediate independently: the second hypersimplex $\Delta_{2,2}$ consists of one lattice point, so every scaling gives the same zero-dimensional toric variety and maximum-likelihood degree $1$. This case must be separated because the source's positive-dimensional Euler-characteristic reduction (2) does not apply to that degenerate hypersimplex.

Finally, (11) simultaneously recovers the principal formulas established separately in the source. For a generic scaling, all principal submatrices of order at least two are nonsingular, whereas each singleton has nullity one. Hence

$$\operatorname{MLdeg}(V_d^c)=2^{d-1}-d. \tag{16}$$

If $\operatorname{rank}C=3$, every principal submatrix of order at least three has rank three because every three-element principal minor equals $2c_{ij}c_{ik}c_{jk}\neq0$. The even-rank subsets are then exactly the empty set, all singletons, and all pairs, giving

$$\operatorname{MLdeg}(V_d^c) =1-d+\binom d2 =\binom{d-1}{2}. \tag{17}$$

Thus the conjectural delta-matroid invariance holds in arbitrary dimension, with the explicit principal-nullity formula (11) extending the source's dimension-at-most-six result.