Zhegalkin (also Žegalkin, Gégalkine or Shegalkin) polynomials form one of many possible representations of the operations of Boolean algebra. Introduced by the Russian mathematician Ivan Ivanovich Zhegalkin in 1927, 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. Boolean equivalent Prior to 1927  Boolean algebra had been considered a calculus of logical values with logical operations of conjunction, disjunction, negation, etc . Zhegalkin showed that all Boolean operations could be written as ordinary numeric polynomials, thinking of the logical constants 0 and 1 as integers mod 2. The logical operation of conjunction is realized as the arithmetic operation of multiplication xy, and logical exclusive-or as arithmetic addition mod 2, (written here as x⊕y to avoid confusion with the common use of + as a synonym for inclusive-or ∨). Logical complement ¬x is then derived from 1 and ⊕ as x⊕1. Since ∧ and ¬ form a sufficient basis for the whole of Boolean algebra, meaning that all other logical operations are obtainable as composites of these basic operations, it follows that the polynomials of ordinary algebra can represent all Boolean operations, allowing Boolean reasoning to be performed reliably by appealing to the familiar laws of elementary algebrawithout the distraction of the differences from high school algebra that arise with disjunction in place of addition mod 2. An exampleapplication is the representation of the Boolean 2-out-of-3 threshold or median operation  as the Zhegalkin polynomial xy⊕yz⊕zx , which is 1 when at least two of the variables are 1 and 0 otherwise .
Zhegalkin (also Žegalkin, Gégalkine or Shegalkin) polynomials are a representation of functions in Boolean algebra. Introduced by the Russian mathematician Ivan Ivanovich Zhegalkin in 1927, 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. Boolean equivalent Prior to 1927 , Boolean algebra had been considered a calculus of logical values with logical operations of conjunction, disjunction, negation, and so on . Zhegalkin showed that all Boolean operations could be written as ordinary numeric polynomials, representing the false and true values as 0 and 1 , the integers mod 2. Logical conjunction is written as xy, and logical exclusive-or as arithmetic addition mod 2, (written here as x⊕y to avoid confusion with the common use of + as a synonym for inclusive-or ∨). The logical complement ¬x is then x⊕1. Since ∧ and ¬ form a basis for Boolean algebra, all other logical operations are compositions of these basic operations, and so the polynomials of ordinary algebra can represent all Boolean operations, allowing Boolean reasoning to be performed using elementary algebra. For example, the Boolean 2-out-of-3 threshold or median operation is written as the Zhegalkin polynomial xy⊕yz⊕zx .
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