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 {\bf  Andr\'e Bouchet and   Bill Jackson }
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\noindent {\bf Parity Systems and the Delta-Matroid Intersection
Problem      }
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\noindent We consider the problem of determining when two delta-matroids on
the
  same ground-set have a common base. Our approach is to adapt the theory
  of matchings in 2-polymatroids developed by Lov\'asz
   $[13]$ to a new abstract system,  which we call  a
  parity system. Examples of parity systems may be obtained by combining
  either, two delta-matroids, or two orthogonal 2-polymatroids, on the
  same ground-sets. We show that many of the results of Lov\'asz
  concerning `double flowers' and `projections' carry over to parity
  systems.


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