Deciding multiaffinity of polynomials over a finite field
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Svetlana N. Selezneva
Abstract
We consider polynomials f(x1, …, xn) over a finite filed that satisfy the following condition: the set of solutions of the equation f(x1, …, xn) = b, where b is some element of the field, coincides with the set of solutions of some system of linear equations over this field. Such polynomials are said to be multiaffine with the right-hand side b (or with respect to b). We describe a number of properties of multiaffine polynomials. Then on the basis of these properties we propose a polynomial algorithm that takes a polynomial over a finite field and an element of the field as an input and decides whether the polynomial is multiaffine with respect to this element. In case of the positive answer the algorithm also outputs a system of linear equations that corresponds to this polynomial. The complexity of the proposed algorithm is the smallest in comparison with other known algorithms that solve this problem.
Originally published in Diskretnaya Matematika (2023) 35, №2, 109–124 (in Russian).
Funding statement: Research was supported by Ministry of Science and Higher Education of the Russian Federation within the framework of the program of Moscow Center of Fundamental and Applied Mathematics under the Agreement 075-15-2022-284.
References
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- Asymptotic local lower deviations of strictly supercritical branching process in a random environment with geometric distributions of descendants
- Cloning operations and graph diameter
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Articles in the same Issue
- Frontmatter
- On large deviations of the moment of attaining far level by the random walk in a random environment
- Asymptotic local lower deviations of strictly supercritical branching process in a random environment with geometric distributions of descendants
- Cloning operations and graph diameter
- Relations between energy complexity measures of Boolean networks and positive sensitivity of Boolean functions
- On the maximal size of tree in a random forest
- Deciding multiaffinity of polynomials over a finite field
- Probability that given vertices belong to the same connected component of random equiprobable mapping
