Zhegalkin polynomial

Zhegalkin (also Žegalkin, Gégalkine or Shegalkin^[1]) polynomials (Russian: полиномы Жегалкина), also known as algebraic normal form, are a representation of functions in Boolean algebra. Introduced by the Russian mathematician Ivan Ivanovich Zhegalkin in 1927,^[2] they are the polynomial ring over the integers modulo 2. The resulting degeneracies of modular arithmetic result in Zhegalkin polynomials being simpler than ordinary polynomials, requiring neither coefficients nor exponents. Coefficients are redundant because 1 is the only nonzero coefficient. Exponents are redundant because in arithmetic mod 2, x² = x. Hence a polynomial such as 3x²y⁵z is congruent to, and can therefore be rewritten as, xyz.

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