Affine-Slice Nonvanishing for non-brace bipartite graphs

Establish that for every bipartite graph G (not necessarily a brace) with edge-coloring ρ and target t, the exact-t Vandermonde polynomial P_{G,t}(λ) = [x^t] det M_G(x,λ) is not identically zero whenever the exact-t fiber F_{G,t} of perfect matchings with exactly t red edges is nonempty; equivalently, show that for any tight-cut decomposition of G into blocks G1 and G2, the convolution sum P_{G,t}(λ) = Σ_{t1+t2=t} P_{G1,t1}(λ)·P_{G2,t2}(λ) cannot cancel to zero whenever F_{G,t} ≠ ∅.

Background

The paper proves the Affine-Slice Nonvanishing Conjecture (ASNC) for all bipartite braces and derives a deterministic O(n6) algorithm for the Exact Matching decision problem on general bipartite graphs. ASNC asserts that the exact-t Vandermonde polynomial is nonzero whenever the exact-t matching fiber is nonempty.

For general (non-brace) bipartite graphs, tight-cut decomposition shows that matchings decompose across blocks, but the corresponding polynomial P_{G,t} is a convolution sum over block polynomials. The authors note that, unlike the decision problem, this sum could cancel even when block polynomials are individually nonzero, and they explicitly state that proving noncancellation (i.e., extending ASNC beyond braces) remains open.

References

Important distinction. Step 1 establishes only the decision reduction: $F_{G,t} \neq \varnothing$ iff there exist $t_1 + t_2 = t$ with both block fibers nonempty. It does not establish polynomial noncancellation for non-braces: the convolution $P_{G,t} = \sum_{t_1+t_2=t} P_{G_1,t_1} \cdot P_{G_2,t_2}$ could in principle cancel even when individual block polynomials are nonzero. Proving ASNC for non-brace graphs (i.e., that this convolution sum is nonzero whenever $F_{G,t} \neq \varnothing$) remains open. The algorithm does not require it: it recurses on blocks and checks each independently.

— Bipartite Exact Matching in P  (2604.01571 - Du, 2 Apr 2026) in Proof of Theorem “Reduction” (Section 3), Step 1: Important distinction