Computer algebra is a diffuse branch of mathematics done with computers that encompasses both computational algebra and analysis.

Computer algebra includes exact symbolic manipulation of mathematical expressions, such as polynomial computation, symbolic integration, symbolic summation, and equation solving. Practical symbolic integration systems combine decision procedures such as the Risch algorithm with heuristic, table-based, and rule-based methods; even when an elementary antiderivative exists, implementations may miss it (Desmond 2026).

Computer algebra can also supply exact checks for results proposed in AI-assisted mathematics. For example, a claimed antiderivative can be tested by differentiation and simplification to determine whether it reproduces the integrand. Such a check verifies the resulting identity, not the correctness or significance of the process that produced the candidate (Wolfram 2026).

Formal proof checking additionally verifies the inference steps and assumptions of a complete derivation. Symbolic calculation can support such an argument, but simplifying an expression or testing a numerical example does not check the complete derivation.

See also

AI-Assisted Mathematics, Algebra, Analysis, Experimental Mathematics, Risch Algorithm, Symbolic Integration

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References

Boulier, F.; Mou, C.; Sadykov, T. M.; and Uncu, A. K. (Eds.). Computer Algebra in Scientific Computing: CASC 2026 Proceedings. Cham, Switzerland: Springer, 2026. https://doi.org/10.1007/978-3-032-34586-8.Cohen, J. S. Computer Algebra and Symbolic Computation: Mathematical Methods. Wellesley, MA: A K Peters, 2003.Davenport, J. H.; Siret, Y.; and Tournier, E. Computer Algebra: Systems and Algorithms for Algebraic Computation, 2nd ed. San Diego, CA: Academic Press, 1993.Desmond, H. "Exhaustive Symbolic Integration: Integration by Differentiation and the Landscape of Symbolic Integrability." 6 May 2026. https://arxiv.org/abs/2605.04978.Gathen, J. von zur and Gerhard, J. Modern Computer Algebra. Cambridge, England: Cambridge University Press, 1999.Geddes, K. O.; Czapor, S. R.; and Labahn, G. Algorithms for Computer Algebra. Amsterdam, Netherlands: Kluwer, 1992.Mishra, B. Algorithmic Algebra. New York: Springer-Verlag, 1993.OpenAI. "Sharing AI Progress in Mathematics." Oct. 6, 2026. https://openai.com/index/sharing-ai-progress-in-mathematics/.Wolfram, S. "What's the Future for Pure Math Research in the Age of AI?" Sep. 28, 2026. https://writings.stephenwolfram.com/2026/09/whats-the-future-for-pure-math-research-in-the-age-of-ai/.

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Computer Algebra

Cite this as:

Weisstein, Eric W. "Computer Algebra." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ComputerAlgebra.html

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