Algorithms for Polynomial Computation and Applications

The computational manipulation of polynomials is a foundational element across pure and applied mathematics, computer algebra, cryptography and scientific computing. Central tasks include multiplication, division and inversion of dense and sparse polynomials, factorisation, multi-point evaluation, interpolation and power-series manipulation. Advances in algorithmic design have steadily reduced asymptotic complexities, achieving quasi-linear time bounds for multiplication via fast Fourier transform analogues, subquadratic division and series reversion through Newton-iteration frameworks, and sparse-direct techniques that exploit term structure to bypass worst-case dense costs. Modern developments integrate modular methods, Chinese remainder reconstructions and linear algebra on structured matrices to handle large degrees and high dimensions. Hardware accelerators, notably GPUs, are exploited to scale operations for cryptographic applications such as fully homomorphic encryption and zero-knowledge proofs. These algorithmic innovations underpin practical libraries and software systems, enabling real-time control computations, symbolic solutions in algebraic geometry and high-throughput data-driven simulations.

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Algorithms for Polynomial Computation and Applications publication trend

The graph below shows the total number of articles in algorithms for polynomial computation and applications across all publications each year (not limited to Nature Index journals).

Technical terms

Number theoretic transform (NTT): A finite-field analogue of the fast Fourier transform used to perform convolution-based polynomial multiplication in quasi-linear time.

Power series division: The process of computing the quotient of two formal power series up to a finite truncation order, often via Newton iteration or direct combinatorial formulas.

Sparse polynomial: A polynomial characterised by having significantly fewer nonzero terms than its total degree, allowing specialised algorithms that leverage term-wise structure.

Tower arithmetic: Arithmetic operations performed in a sequence of nested algebraic field extensions, enabling computations in higher-degree composite fields with structured basis conversions.

References

1. High-Performance Number Theoretic Transform on GPU Through radix2-CT and 4-Step Algorithms. IEEE Access (2025).
2. Accelerated tower arithmetic. Journal of Complexity (2019).
3. A fast algorithm for reversion of power series. Mathematics of Computation (2014).
4. Division of Power Series: Recursive and Non-Recursive Formulas. Anais da Academia Brasileira de Ciências (2022).