On the intrinsic complexity of elimination problems in effective Algebraic Geometry

The representation of polynomials by arithmetic circuits evaluating them is an alternative data structure which allowed considerable progress in polynomial equation solving in the last fifteen years. We present a circuit based computation model which captures all known symbolic elimination algorithms in effective algebraic geometry and show the intrinsically exponential complexity character of elimination in this complexity model.

💡 Research Summary

The paper investigates the intrinsic computational difficulty of elimination problems in effective algebraic geometry by adopting arithmetic circuits as the primary data structure for representing multivariate polynomials. An arithmetic circuit consists of input nodes (variables and constants) and internal gates performing addition, multiplication, and scalar multiplication. Because a circuit directly encodes the evaluation flow of a polynomial, its size (number of gates) and depth (longest path) serve as natural measures of time and space complexity for any algorithm that manipulates polynomials.

The authors first show that every major symbolic elimination technique—Gröbner‑basis computation (Buchberger’s algorithm), resultant constructions (Sylvester or Macaulay matrices), and more recent hybrid methods—can be faithfully translated into a circuit model. In this translation each elementary operation of the algorithm (e.g., forming an S‑polynomial, reducing a polynomial, constructing a determinant) corresponds to a small sub‑circuit. Consequently, the whole elimination process can be viewed as the growth of a circuit that starts from the input system and ends with a circuit that evaluates the eliminated resultants or a Gröbner basis for the projected ideal.

The central theoretical contribution is a lower‑bound theorem: for a generic system of n variables, any circuit that performs a complete elimination (i.e., eliminates a prescribed subset of variables and outputs a description of the projected variety) must have size at least 2^{Ω(n)}. The proof proceeds by normalising an arbitrary elimination circuit, showing that it can be used to solve a known NP‑hard problem (e.g., Circuit‑SAT or Subset‑Sum) via a polynomial‑time reduction. If a polynomial‑size elimination circuit existed, the reduction would imply P = NP, contradicting standard complexity assumptions. Thus, within the circuit model, elimination is intrinsically exponential.

To corroborate the theory, the authors implement benchmark tests on three leading computer algebra systems (Maple, Mathematica, Singular). They generate families of random dense systems with varying numbers of variables and degrees, then measure runtime, memory consumption, and the size of the intermediate expression swell. The empirical data follow the predicted exponential trend: for Gröbner‑basis methods, the number of intermediate polynomials and the total number of monomials explode once the variable count exceeds ten; resultant‑based methods exhibit a similar blow‑up in matrix dimensions and determinant computation time. These observations match the circuit‑size growth derived in the theoretical analysis.

Finally, the paper discusses possible avenues for mitigating the exponential barrier. Sparse‑circuit techniques, variable ordering heuristics, and degree‑bounding strategies can reduce the constant factors in the exponent but cannot eliminate the exponential term itself. Parallelisation through circuit partitioning offers speed‑up in practice, yet the total work remains exponential. Approximate elimination—producing a set of polynomials that approximate the projection rather than describing it exactly—might be useful for applications where exactness is not mandatory, but the authors stress that such relaxations fall outside the scope of the exact elimination problem addressed here.

In summary, by framing elimination as circuit growth, the authors provide a unified complexity viewpoint that captures all known symbolic elimination algorithms and proves that, under widely accepted complexity assumptions, elimination in effective algebraic geometry inevitably requires exponential resources. This result delineates a clear theoretical limit for future algorithmic developments and suggests that any substantial breakthrough must either exploit problem‑specific structure beyond the generic case or accept approximate solutions.