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- W1487208937 abstract "AbstractWegiveanexamplewherethenumberofelementsofaGrobnerbasisinaBooleanringisnotpolynomiallyboundedintermsofthebitsizeanddegreesoftheinput. 1 Boolean Rings Let R n = F 2 [x 1 ,...,x n ,y 1 ,...,y n ,z 1 ,...,z n ] where F 2 = Z/(2). If S ⊆ R n then (S) denotes the ideal in R n generated by S, and Sol(S) ⊆ K 3n denotes thesolution set of S, where K is the algebraic closure of F 2 . LetS n := {c 2 −c| c∈ {x 1 ,...,x n ,y 1 ,...,y n ,z 1 ,...,z n }}.R bn := R n /(S n ) is a Booleanring , which means r 2 = rfor all rin R bn . If I is anideal in R n , then S n ⊆ I if and only if I is radical and Sol(I) ⊆ F 3n2 .Ideals in R n that contain S n are in 1-1 correspondence with ideals in R bn .Thus, Gr¨obner basis in R bn is equivalent to: Gr¨obner basis in R restrictedto ideals that contain S n . We will show with an example that this is notpolynomial-space. 2 Example LetL n = {x i y i +x i +y i −z i | i= 1,...,n}T n = {x i z i −x i | i= 1,...,n}[{y i z i −y i | i= 1,...,n}P n = {Y ni=1 c i | c" @default.
- W1487208937 created "2016-06-24" @default.
- W1487208937 creator A5045195686 @default.
- W1487208937 date "2015-02-25" @default.
- W1487208937 modified "2023-09-27" @default.
- W1487208937 title "Groebner basis in Boolean rings is not polynomial-space" @default.
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