Subtraction-free characterization of bounded ratios

Prove that every integral exponent vector whose ratio is bounded on products of positive linear forms in n variables and k variables defines a subtraction-free ratio; equivalently, establish that if d4a4d4e belongs to the bounded-ratio cone for products of linear forms, then the polynomial C D_d4a4 - N_d4a4 has only nonnegative coefficients for some positive constant C, where N_d4a4 and D_d4a4 are the numerator and denominator monomials of the ratio.

Background

The paper defines a ratio of coefficients of a product of n linear forms as subtraction-free when, after writing the ratio as N_\gamma/D_\gamma, the polynomial C D_\gamma-N_\gamma has only nonnegative coefficients in the linear-form parameters. Any such certificate immediately proves boundedness with optimal bound at most C.

Computations in the paper show that the conjectured property holds with C=1 for all nine extreme rays of the bounded-ratio cone for products of three linear forms in three variables and for all eighty extreme rays in the case of three linear forms in four variables. The conjecture generalizes an earlier conjecture for degree two and asks whether boundedness is always witnessed by a subtraction-free polynomial certificate.

References

Conjecture~\ref{conj:sf} extends to arbitrary n the corresponding conjecture of Conjecture~5.5 for n = 2. That conjecture is stated in terms of the normalized coefficients $!\,P_{}$. The required constant is the power of two $2{\sum_i \gamma_{2e_i}$, which is exactly the factorial correction $\prod_{} (!){\gamma_{}$. Indeed, in degree two, $! = 2$ for $ = 2e_i$ and $! = 1$ otherwise. In degree $n$, the numbers $! = \alpha_1! \cdots \alpha_k!$ are no longer powers of two, so no such constant is available. The correction is absorbed into the coefficients $P_{}$ used here. This is why Definition~\ref{def:sf} allows an unspecified constant $C$.

\begin{conjecture}\label{conj:sf} If $ \in BR_{\mathring{Q}(n,k) \cap \mathbb{Z}{H(n,k)}$, then $R_{}$ is subtraction-free. \end{conjecture}

— Bounded ratios for Lorentzian polynomials  (2609.05341 - Bathija et al., 4 Sep 2026) in Concluding remarks, Definition of subtraction-free ratio, Conjecture~\ref{conj:sf}