Lifts of matroid representations over partial fields

There exist several theorems which state that when a matroid is representable over distinct fields F 1 , … , F k , it is also representable over other fields. We prove a theorem, the Lift Theorem, that implies many of these results. parts of Whittleʹs characterization of representations of ternary matroids follow from our theorem. Second, we prove the following theorem by Vertigan: if a matroid is representable over both GF ( 4 ) and GF ( 5 ) , then it is representable over the real numbers by a matrix such that the absolute value of the determinant of every nonsingular square submatrix is a power of the golden ratio. Third, we give a characterization of the 3-connected matroids having at least two inequivalent representations over GF ( 5 ) . We show that these are representable over the complex numbers. onally we provide an algebraic construction that, for any set of fields F 1 , … , F k , gives the best possible result that can be proven using the Lift Theorem.