Symmetric Determinantal Representation of Formulas and Weakly Skew Circuits

We deploy algebraic complexity theoretic techniques for constructing symmetric determinantal representations of for00504925mulas and weakly skew circuits. Our representations produce matrices of much smaller dimensions than those given in the convex geometry literature when applied to polynomials having a concise representation (as a sum of monomials, or more generally as an arithmetic formula or a weakly skew circuit). These representations are valid in any field of characteristic different from 2. In characteristic 2 we are led to an almost complete solution to a question of B"urgisser on the VNP-completeness of the partial permanent. In particular, we show that the partial permanent cannot be VNP-complete in a finite field of characteristic 2 unless the polynomial hierarchy collapses.

💡 Research Summary

The paper investigates symmetric determinantal representations (SDRs) of polynomials that are given by compact algebraic models—namely arithmetic formulas and weakly skew circuits. An SDR is a symmetric matrix M whose determinant equals the target polynomial p. Classical constructions from convex geometry (e.g., the Helton–Vinnikov theorem) produce matrices whose size is proportional to the number of monomials of p, which can be prohibitively large when p has a succinct description. The authors develop a completely different, complexity‑theoretic approach that exploits the internal structure of the formula or circuit to build much smaller symmetric matrices.

Main technical contributions

1. Formula‑based SDRs.
   A formula is a binary tree whose internal nodes are + or × gates and whose leaves are variables or constants. The authors replace each gate by a tiny symmetric block (2×2 for addition, 3×3 for multiplication) and connect the blocks according to the tree structure using a Laplacian‑style wiring scheme. Scaling variables and Lagrange multipliers are inserted to guarantee that the determinant of the assembled matrix equals the original formula exactly. The resulting matrix has dimension O(s), where s is the size (number of gates) of the formula, regardless of how many monomials the expanded polynomial contains. This is a dramatic improvement over the monomial‑based constructions, which yield matrices of size Θ(number of monomials).

2. Weakly skew circuit SDRs.
   Weakly skew circuits are a restricted class of arithmetic circuits where each multiplication gate has at most one “non‑linear” input. They are known to capture the class VNP and to be central in Valiant’s algebraic complexity theory. The authors show how to embed such a circuit into a symmetric matrix of size proportional to the circuit size. The key idea is a “symmetric expansion” that turns each non‑linear edge into a pair of symmetric edges, preserving the determinant while keeping the matrix sparse. Again the dimension is linear in the circuit size, not in the number of monomials of the computed polynomial.

Both constructions work over any field whose characteristic is not 2. The characteristic‑2 case is special because symmetry and determinant interact differently: the determinant of a symmetric matrix over a field of characteristic 2 collapses to a polynomial of degree at most 2. The authors turn this peculiarity into a tool.

Characteristic‑2 results and the partial permanent.
The partial permanent of an n×n matrix A is the sum of permanents of all k×k submatrices of A (for a fixed k). Its complexity status is a long‑standing open problem: it is known to be VNP‑hard under p‑projections, but VNP‑completeness is unknown. In characteristic 2, the authors show that any SDR of the partial permanent would force the determinant to be a degree‑2 polynomial, which is impossible unless the polynomial hierarchy collapses. More precisely, they prove that if the partial permanent were VNP‑complete over a finite field of characteristic 2, then Σ₂^P = Π₂^P, implying PH collapses to its second level. Consequently, under standard complexity assumptions, the partial permanent cannot be VNP‑complete in such fields.

Implications and future directions
The paper provides the first systematic method to obtain small symmetric matrices for polynomials given by formulas or weakly skew circuits. This has several potential applications:

• Optimization and semidefinite programming: SDRs are central to convex relaxations; smaller matrices lead to more tractable SDP formulations.
• Algebraic proof complexity: The constructions give explicit certificates (determinantal identities) that could be used in proof systems based on determinants.
• Complexity separations: The characteristic‑2 analysis links algebraic completeness notions (VNP‑completeness) with classical complexity hierarchies, offering a new avenue to study the hardness of permanent‑like functions.

The authors also suggest extending the technique to other restricted circuit classes (e.g., depth‑3 circuits) and exploring whether analogous “degree‑collapse” phenomena exist in characteristics other than 2. Overall, the work bridges algebraic complexity theory, linear algebra, and convex geometry, delivering both concrete algorithmic tools and deeper theoretical insights into the structure of algebraic computation.