Algebra I 18ed by Armando Rojo

By Armando Rojo

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We notice that if the degree ordering is used, this strategy seems to lead to simpler polynomials than other AIMS Essay 2006. Mohamed Saeed Taha. The Complexity of Groebner Bases 30 choices. The output of the Buchberger algorithm after we apply these criteria is not a reduced Groebner basis; there may be redundant polynomials that can be eliminated, yet G is still a Groebner basis. If LT (gi ) divides LT (g j ) then g j , can be eliminated from the basis G and G {g j } is still a Groebner basis.

1 Nondeterministic Turing Machines A nondeterministic Turing machine is a Turing machine for which, at any point in a computation, the machine may proceed according to several possibilities. The transition function for the nondeterministic Turing machine has the form: δ : Q × Γ −→ P(Q × Γ × {L, R}) AIMS Essay 2006. Mohamed Saeed Taha. The Complexity of Groebner Bases 25 The computation here is a tree whose branches represent the different possibilities of the machine; if one branch leads to the state “accept”, the machine accepts the input.

In other words, P is The class of languages where membership can be decided quickly, and NP represents The class of languages where membership can be verified quickly. One of the greatest unsolved problems in theoretical computer science is to determine whether these two classes are equal. 3 NP-hardness and NP-completeness The problem is NP-hard if any NP problem can be translated into this problem in a polynomial time. e. it is verifiable in a polynomial time in deterministic machine, and any NP problem can be translated to it in polynomial time.