Lower bounds for contingency tables via Lorentzian polynomials

Signed permutohedra, delta‐matroids, and beyond

“…By [21, Lemma 4.8], this is indeed a strengthening of the statement that () is denormalized Lorentzian. Even when $$\mathrm{D}$$ has an enveloping matroid, we do not know if there is a denormalized Lorentzian evaluation of the multivariable $$U$$‐polynomial that specializes to ().…”

“…Proof To obtain the first two results, we set $$x = 0, q = 1$$ in () and set $$x = y = 0, w=1$$ in (), respectively, and then apply [21, Lemma 4.8]. To obtain the last two results, we normalize () and set $$y = q = 1/2, x_i = u$$ and set $$y = 0, q = 1, x_i = u$$, respectively, and then apply [20, Corollary 3.7].$$\Box$$…”

“…Then the polynomials $$$$\begin{equation} (y + q)^n\, U_{\mathrm{D}}{\left(\frac{x}{y + q}, \frac{y - q}{y + q}\right)} \quad \text{and } \end{equation}$$$$ $$$$\begin{equation} (y + w)^n\, U_{\mathrm{D}}{\left(\frac{2z + x}{y + w}, \frac{y - z}{y + w}\right)} \end{equation}$$$$have a log‐concave unbroken array of coefficients. In fact, they are denormalized Lorentzian polynomials in the sense of [20, 21].…”

Stellahedral geometry of matroids

“…Lorentzian polynomials satisfy a strong log-concavity property [BH20, Example 2.26] and are preserved under nonnegative linear change of variables [BH20, Theorem 2.10]. Polynomials whose normalization is Lorentzian, called denormalized Lorentzian polynomials , share similar properties [BLP23, §4.3].…”

“…For any rank r matroid on E , the polynomial is a denormalized Lorentzian polynomial in the sense of [BH20; BLP23].…”