Zhegalkin polynomial

Zhegalkin (also Zegalkin or Gegalkine) polynomials form one of many possible representations of the operations of Boolean algebra. Introduced by the Russian mathematician I.I. Zhegalkin in 1927, they are the polynomials of ordinary high school algebra interpreted over the integers mod 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 truth values with logical operations of conjunction, disjunction, negation, etc. Zhegalkin showed that all Boolean operations could be written as ordinary numeric polynomials. Logical negation &not;x is realized as the Zhegalkin polynomial x&oplus;1 and logical conjunction as multiplication. (In the context of Boolean algebra the addition operation mod 2, doubling as the logical operation exclusive-or, is written &oplus; in preference to + to avoid confusion with the latter's occasional use in place of &or; to denote disjunction or inclusive-or in Boolean algebra.) These two logical operations form a sufficient basis for the whole of Boolean algebra, meaning that all other logical operations are obtainable as composites of these basic operations. The polynomials of ordinary algebra then represent all Boolean operations, allowing Boolean reasoning to be performed reliably by appealing to the familiar laws of high school algebra without the distraction of the differences from high school algebra that arise with disjunction in place of addition mod 2.

An example application is the representation of the Boolean 2-out-of-3 threshold or median operation as the Zhegalkin polynomial xy&oplus;yz&oplus;zx, which is 1 when at least two of the variables are 1 and 0 otherwise.

Formal properties
Formally a Zhegalkin monomial is the product of a finite set of variables (hence square-free), including the empty set whose product is denoted 1. There are 2n possible Zhegalkin monomials in n variables, since each monomial is fully specified by the presence or absence of each variable. A Zhegalkin polynomial is the sum (exclusive-or) of a set of Zhegalkin monomials, with the empty set denoted by 0. A given monomial's presence or absence in a polynomial corresponds to that monomial's coefficient being 1 or 0 respectively. The Zhegalkin monomials, being linearly independent, span a 2n-dimensional vector space over the Galois field GF(2) (NB: not GF(2n) whose multiplication is quite different). The 22 n vectors of this space, as the linear combinations of those monomials as unit vectors, constitute the Zhegalkin polynomials. The exact agreement with the number of Boolean operations on n variables, which exhaust the n-ary operations on {0,1}, furnishes a direct counting argument for completeness of the Zhegalkin polynomials as a Boolean basis.

This vector space is not equivalent to the free Boolean algebra on n generators because it lacks complementation (bitwise logical negation) as an operation (equivalently, because it lacks top as a constant). This is not to say that the space is not closed under complementation or lacks top (the all-ones vector) as an element, but rather that the linear transformations of this and similarly constructed spaces need not preserve that operation and element. Those that do correspond to the Boolean homomorphisms, e.g. there are four linear transformations from the vector space of Zhegalkin polynomials over one variable to that over none, only two of which are Boolean homomorphisms.

Related work
In the same year as Zhegalkin's paper (1927) the American mathematician E.T. Bell published a sophisticated arithmetization of Boolean algebra based on Dedekind's ideal theory and general modular arithmetic (as opposed to arithmetic mod 2). The much simpler arithmetic character of Zhegalkin polynomials was first noticed in the west (independently, communication between Soviet and western mathematicians being very limited in that era) by the American mathematician Marshall Stone in 1936 when he observed while writing up his celebrated Stone duality theorem that the supposedly loose analogy between Boolean algebras and rings could in fact be formulated as an exact equivalence holding for both finite and infinite algebras, leading him to substantially reorganize his paper.