Generalizing the Multiple Exchange Property for Matroid Bases

The multiple exchange property for matroid bases states that for any bases $A$ and $B$ of a matroid and any subset $X\subseteq A\setminus B$, there exists a subset $Y\subseteq B\setminus A$ such that both $A-X+Y$ and $B+X-Y$ are bases. This classical result has found applications not only in matroid theory, but also in the analysis and design of various algorithms. This paper generalizes the multiple exchange property in two directions. First, we prove a common generalization of this and other known basis exchange properties by showing that for any subsets $X \subseteq A \setminus B$ and $Y \subseteq B \setminus A$, there exist subsets $U \subseteq A \setminus B$ and $V \subseteq B \setminus A$ such that $X\subseteq U$, $Y\subseteq V$, $A-U+V$ and $B+U-V$ are bases, and $|U|=|V|$ is at most the rank of $X+Y$. Second, we develop a general framework for deriving extensions of the Grassmann-Pl\"ucker identity, showing further exchange properties for matroids representable over a field of characteristic zero. Using our framework, we prove an exchange property for this matroid class that simultaneously generalizes our first result and the very recent Equitability Theorem (SODA 2026).