Multiaffine polynomials over a finite field
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Svetlana N. Selezneva
Abstract
We consider polynomials f(x1, …, xn) over a finite field that possess the following property: for some element b of the field the set of solutions of the equation f(x1, …, xn) = b coincides with the set of solutions of some system of linear equations over this field. Such polynomials are said to be multiaffine with respect to the right-hand side b. We obtain the properties of multiaffine polynomials over a finite field. Then we show that checking the multiaffinity with respect to a given right-hand side may be done by an algorithm with polynomial (in terms of the number of variables and summands of the input polynomial) complexity. Besides that, we prove that in case of the positive decision a corresponding system of linear equations may be recovered with complexity which is also polynomial in terms of the same parameters.
Note: Originally published in Diskretnaya Matematika (2020) 32,№3, 85–97 (in Russian).
Acknowledgment
The author thanks the anonymous referee for useful comments and questions that made it possible to improve the presentation of results.
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Funding: Research was supported by RBRF, project 19-01-00200-a.
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© 2021 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
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- Ergodicity of the probabilistic converter, a serial connection of two automata
- On distance-regular graphs with c2 = 2
- Minimal contact circuits for characteristic functions of spheres
- Approximation of restrictions of q-valued logic functions to linear manifolds by affine analogues
- Multiaffine polynomials over a finite field
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Articles in the same Issue
- Frontmatter
- A method of construction of differentially 4-uniform permutations over Vm for even m
- Ergodicity of the probabilistic converter, a serial connection of two automata
- On distance-regular graphs with c2 = 2
- Minimal contact circuits for characteristic functions of spheres
- Approximation of restrictions of q-valued logic functions to linear manifolds by affine analogues
- Multiaffine polynomials over a finite field
- Large deviations of branching process in a random environment. II
