We study the decision problem Affine Rank Minimization, denoted ARM(k). The input consists of rational matrices A_1,...,A_q in Q^{m x n} and rational scalars b_1,...,b_q in Q. The question is whether there exists a real matrix X in R^{m x n} such that trace(A_l^T X) = b_l for all l in {1,...,q} and rank(X) <= k. We first prove membership: for every fixed k >= 1, ARM(k) lies in the existential theory of the reals by giving an explicit existential encoding of the rank constraint using a constant-size factorization witness. We then prove existential-theory-of-reals hardness via a polynomial-time many-one reduction from ETR to ARM(k), where the target instance uses only affine equalities together with a single global constraint rank(X) <= k. The reduction compiles an ETR formula into an arithmetic circuit in gate-equality normal form and assigns each circuit quantity to a designated entry of X. Affine semantics (constants, copies, addition, and negation) are enforced by linear constraints, while multiplicative semantics are enforced by constant-size rank-forcing gadgets. Soundness is certified by a fixed-rank gauge submatrix that removes factorization ambiguity. We prove a composition lemma showing that gadgets can be embedded without unintended interactions, yielding global soundness and completeness while preserving polynomial bounds on dimension and bit-length. Consequently, ARM(k) is complete for the existential theory of the reals; in particular, ARM(3) is complete. This shows that feasibility of purely affine constraints under a fixed constant rank bound captures the full expressive power of real algebraic feasibility.

翻译：我们研究决策问题仿射秩最小化，记作ARM(k)。输入包含有理矩阵A_1,...,A_q ∈ Q^{m×n}与有理标量b_1,...,b_q ∈ Q。问题在于是否存在实数矩阵X ∈ R^{m×n}使得对所有l∈{1,...,q}满足trace(A_l^T X) = b_l且秩rank(X) ≤ k。我们首先证明成员关系：对于任意固定k≥1，ARM(k)属于实数存在理论，通过使用常数规模的分解见证显式编码秩约束。随后通过从ETR到ARM(k)的多项式时间多一归约证明实数存在理论的困难性，其中目标实例仅使用仿射等式与单一全局约束rank(X) ≤ k。该归约将ETR公式编译为门等式正规形式的算术电路，并将每个电路量分配到X的指定元素中。仿射语义（常数、复制、加法与取负）通过线性约束实现，而乘法语义通过常数规模的秩强制构件实现。可靠性由固定秩规范子矩阵认证，该子矩阵消除了分解歧义。我们证明了一个组合引理，表明构件可被嵌入且不会产生非预期交互，从而在保持维度与比特长度多项式界限的同时实现全局可靠性与完备性。因此，ARM(k)对实数存在理论是完全的；特别地，ARM(3)是完全的。这表明在固定常数秩界下纯仿射约束的可行性捕捉了实代数可行性的完整表达能力。