For a real matrix $A \in \mathbb{R}^{d \times n}$ with non-collinear columns, we show that $n \leq O(d^4 \kappa_A)$ where $\kappa_A$ is the \emph{circuit imbalance measure} of $A$. The circuit imbalance measure $\kappa$ is a real analogue of $\Delta$-modularity for integer matrices, satisfying $\kappa_A \leq \Delta_A$ for integer $A$. The circuit imbalance measure has numerous applications in the context of linear programming (see Ekbatani, Natura and V{\'e}gh (2022) for a survey). Our result generalizes the $O(d^4 \Delta_A)$ bound of Averkov and Schymura (2023) for integer matrices and provides the first polynomial bound holding for all parameter ranges on real matrices. To derive our result, similar to the strategy of Geelen, Nelson and Walsh (2021) for $\Delta$-modular matrices, we show that real representable matroids induced by $\kappa$-bounded matrices are minor closed and exclude a rank $2$ uniform matroid on $O(\kappa)$ elements as a minor (also known as a line of length $O(\kappa)$). As our main technical contribution, we show that any simple rank $d$ complex representable matroid which excludes a line of length $l$ has at most $O(d^4 l)$ elements. This complements the tight bound of $(l-3)\binom{d}{2} + d$ for $l \geq 4$, of Geelen, Nelson and Walsh which holds when the rank $d$ is sufficiently large compared to $l$ (at least doubly exponential in $l$).

翻译：对于列向量非共线的实矩阵 $A \in \mathbb{R}^{d \times n}$，我们证明 $n \leq O(d^4 \kappa_A)$ 成立，其中 $\kappa_A$ 是 $A$ 的\emph{电路不平衡度}。电路不平衡度 $\kappa$ 是整数矩阵 $\Delta$-模性的实数推广，对整数矩阵 $A$ 满足 $\kappa_A \leq \Delta_A$。电路不平衡度在线性规划领域具有广泛的应用（详见 Ekbatani、Natura 和 V{\'e}gh（2022）的综述）。我们的结果推广了 Averkov 和 Schymura（2023）对整数矩阵建立的 $O(d^4 \Delta_A)$ 界，并为实矩阵的所有参数范围提供了首个多项式界。为推导此结果，我们借鉴 Geelen、Nelson 和 Walsh（2021）处理 $\Delta$-模矩阵的策略，证明由 $\kappa$-有界矩阵诱导的实可表示拟阵是子式封闭的，且排除一个在 $O(\kappa)$ 个元素上的秩 $2$ 均匀拟阵作为子式（亦称为长度为 $O(\kappa)$ 的线）。作为主要技术贡献，我们证明任何排除长度为 $l$ 的线的简单秩 $d$ 复可表示拟阵至多包含 $O(d^4 l)$ 个元素。这补充了 Geelen、Nelson 和 Walsh 建立的紧界 $(l-3)\binom{d}{2} + d$（当 $l \geq 4$ 时），该界在秩 $d$ 相对于 $l$ 充分大（至少为 $l$ 的双指数级）时成立。