  We introduce determinantal sieving, a new, remarkably powerful tool in the
toolbox of algebraic FPT algorithms. Given a polynomial $P(X)$ on a set of
variables $X=\{x_1,\ldots,x_n\}$ and a linear matroid $M=(X,\mathcal{I})$ of
rank $k$, both over a field $\mathbb{F}$ of characteristic 2, in $2^k$
evaluations we can sieve for those terms in the monomial expansion of $P$ which
are multilinear and whose support is a basis for $M$. Alternatively, using
$2^k$ evaluations of $P$ we can sieve for those monomials whose odd support
spans $M$. Applying this framework, we improve on a range of algebraic FPT
algorithms, such as:
  1. Solving $q$-Matroid Intersection in time $O^*(2^{(q-2)k})$ and $q$-Matroid
Parity in time $O^*(2^{qk})$, improving on $O^*(4^{qk})$ over general fields
(Brand and Pratt, ICALP 2021)
  2. $T$-Cycle, Colourful $(s,t)$-Path, Colourful $(S,T)$-Linkage in undirected
graphs, and the more general Rank $k$ $(S,T)$-Linkage problem, all in
$O^*(2^k)$ time, improving on $O^*(2^{k+|S|})$ and $O^*(2^{|S|+O(k^2
\log(k+|\mathbb{F}|))})$ respectively (Fomin et al., SODA 2023)
  3. Many instances of the Diverse X paradigm, finding a collection of $r$
solutions to a problem with a minimum mutual distance of $d$ in time
$O^*(2^{r(r-1)d/2})$, improving solutions for $k$-Distinct Branchings from time
$2^{O(k \log k)}$ to $O^*(2^k)$ (Bang-Jensen et al., ESA 2021), and for Diverse
Perfect Matchings from $O^*(2^{2^{O(rd)}})$ to $O^*(2^{r^2d/2})$ (Fomin et al.,
STACS 2021)
  Here, all matroids are assumed to be represented over fields of
characteristic 2. Over general fields, we achieve similar results at the cost
of using exponential space by working over the exterior algebra. For a class of
arithmetic circuits we call strongly monotone, this is even achieved without
any loss of running time. However, the odd support sieving result appears to be
specific to working over characteristic 2.
