Value-Space Contractions for Fixed-Diagonal Rank Fibers\\ of Symmetric Matrices in Characteristic Two

Abstract

Let $K$ be a field of characteristic two and let $k=K^2$ be its subfield of squares. For a prescribed diagonal vector $d=(d_1,\ldots,d_n)\in K^n$, we study the symmetric matrices of diagonal $d$, graded by rank. The associated quasilinear quadratic form $q_d(x)=\sum_{i=1}^n d_i x_i^2$ has the classical represented-value space $U(d)=D(q_d)=\Span_k\{d_1,\ldots,d_n\}\subseteq K.$ Writing $\omega(d)=\dim_k U(d)$, we construct an explicit, choice-dependent Schur-complement bijection that removes $\omega(d)$ coordinates, shifts rank by $\omega(d)$, and leaves a smaller symmetric matrix whose diagonal entries lie in $U(d)$. From this contraction we determine every possible rank in a fixed-diagonal fiber. We also distinguish two exact classification levels: congruence-induced equivalence is classified by the embedded subspace $U(d)\subseteq K$, whereas abstract rank-preserving bijection type is classified by the integer $\omega(d)$. A separate zero-diagonal bijection relates alternating matrices of size $n$ to symmetric matrices of size $n-1$ with square-valued diagonal and yields a rank rule valid over every field of characteristic two. Over perfect fields the statements specialize to one-coordinate contractions and recover known finite-field rank recurrences. Represented-value spaces and the finite-field enumerations are classical. To the best of our knowledge, the explicit arbitrary-field contractions, the complete fixed-diagonal rank spectrum derived from them, and the resulting two-level classification are new.