## Abstract

Though it goes without saying that linear algebra is fundamental to mathematical biology, polynomial algebra is less visible. In this article, we will give a brief tour of four diverse biological problems where multivariate polynomials play a central role—a subfield that is sometimes called *algebraic biology*. Namely, these topics include biochemical reaction networks, Boolean models of gene regulatory networks, algebraic statistics and genomics, and place fields in neuroscience. After that, we will summarize the history of discrete and algebraic structures in mathematical biology, from their early appearances in the late 1960s to the current day. Finally, we will discuss the role of algebraic biology in the modern classroom and curriculum, including resources in the literature and relevant software. Our goal is to make this article widely accessible, reaching the mathematical biologist who knows no algebra, the algebraist who knows no biology, and especially the interested student who is curious about the synergy between these two seemingly unrelated fields.

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## Acknowledgements

The authors would like to thank Elena Dimitrova, Heather Harrington, Reinhard Laubenbacher, and Raina Robeva for their feedback on an earlier draft of this article. The authors are also grateful to two anonymous referees who provided a detailed critiques and suggestions that gave us a fresh perspective led to a number of improvements.

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Matthew Macauley is partially supported by Simons Foundation Grant #358242. Nora Youngs is supported by the Clare Boothe Luce Program.

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Macauley, M., Youngs, N. The Case for Algebraic Biology: from Research to Education.
*Bull Math Biol* **82**, 115 (2020). https://doi.org/10.1007/s11538-020-00789-w

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DOI: https://doi.org/10.1007/s11538-020-00789-w

### Keywords

- Algebraic biology
- Algebraic statistics
- Biochemical reaction network
- Boolean model
- Combinatorial neural code
- Computational algebra
- Finite field
- Gröbner basis
- Mathematics education
- Polynomial algebra
- Pseudomonomial
- Phylogenetic model
- Place field
- Neural ideal