  We present representative sets-style statements for linear delta-matroids,
which are set systems that generalize matroids, with important connections to
matching theory and graph embeddings. Furthermore, our proof uses a new
approach of sieving polynomial families, which generalizes the linear algebra
approach of the representative sets lemma to a setting of bounded-degree
polynomials. The representative sets statements for linear delta-matroids then
follow by analyzing the Pfaffian of the skew-symmetric matrix representing the
delta-matroid. Applying the same framework to the determinant instead of the
Pfaffian recovers the representative sets lemma for linear matroids.
Altogether, this significantly extends the toolbox available for kernelization.
  As an application, we show an exact sparsification result for Mader networks:
Let $G=(V,E)$ be a graph and $\mathcal{T}$ a partition of a set of terminals $T
\subseteq V(G)$, $|T|=k$. A $\mathcal{T}$-path in $G$ is a path with endpoints
in distinct parts of $\mathcal{T}$ and internal vertices disjoint from $T$. In
polynomial time, we can derive a graph $G'=(V',E')$ with $T \subseteq V(G')$,
such that for every subset $S \subseteq T$ there is a packing of
$\mathcal{T}$-paths with endpoints $S$ in $G$ if and only if there is one in
$G'$, and $|V(G')|=O(k^3)$. This generalizes the (undirected version of the)
cut-covering lemma, which corresponds to the case that $\mathcal{T}$ contains
only two blocks.
  To prove the Mader network sparsification result, we furthermore define the
class of Mader delta-matroids, and show that they have linear representations.
This should be of independent interest.
