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DOI: 10.4230/LIPIcs.APPROX/RANDOM.2026.64

Lifting Polynomial Complexity Measures Using Error-Correcting Codes

Authors: Ivan Hu and Dieter van Melkebeek

Published in: LIPIcs, Volume 392, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026)

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.

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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)

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@InProceedings{hu_et_al:LIPIcs.APPROX/RANDOM.2026.64,
  author =	{Hu, Ivan and van Melkebeek, Dieter},
  title =	{{Lifting Polynomial Complexity Measures Using Error-Correcting Codes}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026)},
  pages =	{64:1--64:10},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-449-9},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{392},
  editor =	{Singh, Mohit and Gur, Tom},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2026.64},
  URN =		{urn:nbn:de:0030-drops-277813},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2026.64},
  annote =	{Keywords: lifting, lower bound, error-correcting code, read-once oblivious algebraic branching program, polynomial factorization}
}

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Lifting Polynomial Complexity Measures Using Error-Correcting Codes

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