### Abstract

We formalise the proof of an important theorem in additive combinatorics due to Khovanskii, attesting that the cardinality of the set of all sums of $n$ many elements of $A$, where $A$ is a finite subset of an abelian group, is a polynomial in $n$ for all sufficiently large $n$. We follow a proof due to Nathanson and Ruzsa as presented in the notes “Introduction to Additive Combinatorics” by Timothy Gowers for the University of Cambridge.

### License

BSD License

### Topics

- Mathematics/Combinatorics

### Session Khovanskii_Theorem

- FiniteProduct
- Khovanskii

### Depends on

- Bernoulli Numbers
- A Case Study in Basic Algebra
- The Plünnecke-Ruzsa Inequality

### Auto-related entries

- The Balog–Szemerédi–Gowers Theorem
- Hensel’s Lemma for the p-adic Integers
- Banach-Steinhaus Theorem

## Cite

×

Khovanskii_Theorem-AFP

    @article{Khovanskii_Theorem-AFP,
      author  = {Angeliki Koutsoukou-Argyraki and Lawrence C. Paulson},
      title   = {Khovanskii's Theorem},
      journal = {Archive of Formal Proofs},
      month   = {September},
      year    = {2022},
      note    = {\url{https://isa-afp.org/entries/Khovanskii_Theorem.html},
                 Formal proof development},
      ISSN    = {2150-914x},
    }

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