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Lifting Polynomial Complexity Measures Using Error-Correcting Codes

Authors Ivan Hu, Dieter van Melkebeek

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LIPIcs.APPROX-RANDOM.2026.64.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Algebraic complexity theory
Keywords
  • lifting
  • lower bound
  • error-correcting code
  • read-once oblivious algebraic branching program
  • polynomial factorization

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Abstract

We describe a method that lifts an arbitrary polynomial f with sparsity s to a polynomial that requires a read-once oblivious algebraic program of width s for every variable order. To do so, we introduce a technique for constructing a gadget based on erasure codes over finite fields, where each variable in f is substituted with a monomial that encodes a codeword into the exponents of the monomial. Our construction represents the first use of error-correcting codes in the context of lifting. As an application, we consider factor complexity, which studies how much the complexity of a polynomial can increase under factorization. Our result allows us to lift any gap in sparsity to the same gap in width in a generic manner.

Cite As Get BibTex

Ivan Hu and Dieter van Melkebeek. Lifting Polynomial Complexity Measures Using Error-Correcting Codes. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 392, pp. 64:1-64:10, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2026.64

Author Details

Ivan Hu
  • University of Wisconsin-Madison, WI, USA
Dieter van Melkebeek
  • University of Wisconsin-Madison, WI, USA

Funding

This material is based upon work supported by the National Science Foundation under Grant Nos. 2137424 and 2312540.

Acknowledgements

The authors thank Nutan Limaye for useful discussions, in particular for background on algebraic proof systems, and the anonymous reviewers for helpful suggestions.

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