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Boolean ideals and their varieties. (English) Zbl 1318.13045
This paper is concerned with ideals in $$R = \mathbb{Z}_2[x_1,\dots,x_n]/(x_i^2-x_i, i=1,\dots,n)$$. The ring $$R$$ is a Boolean ring: every element is idempotent. An immediate consequence is that every ideal is principal. For example $$(x_1,x_2) = (x_1x_2 + x_1 + x_2)$$. For a general ideal (as a function of $$n$$) it is however not quick to write down the generator of an ideal in $$R$$ (given several generators of the same ideal). The bottleneck is an explosion in the number of terms. In fact, when it comes to complexity theory, a major source of motivation for the study of $$R$$ is the SAT problem of satisfyability of Boolean clauses.
The main thread of this paper is a study of the relationship between the variety of an ideal $$I\subset R$$ and the exponents of the monomials of the generator of $$I$$. To see the connection, let $$\phi : R\to R$$ be the following map. Given $$f\in R$$, first compute the variety of $$f$$ in $$\mathbb{Z}_2^n$$. Next turn each point in the variety into a monomial, using the point as the exponent vector. Finally, sum the resulting monomials to get $$\phi(f)$$. One main result of this paper is that $$\phi^4$$ is the identity and on the way to this result the author notes many interesting and useful properties of ideals in $$R$$.
The methods are Gröbner-free, but the paper also gives a nice account of the connections to Gröbner-basis theory and discusses complexity theoretic implications.

##### MSC:
 13P15 Solving polynomial systems; resultants 06E20 Ring-theoretic properties of Boolean algebras 06E30 Boolean functions 68W30 Symbolic computation and algebraic computation 12Y05 Computational aspects of field theory and polynomials (MSC2010) 13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)
PolyBoRi
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