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More on Noncommutative Polynomial Identity Testing
"... We continue the study of noncommutative polynomial identity testing initiated by Raz and Shpilka and present efficient algorithms for the following problems in the noncommutative model: Polynomial identity testing: The algorithm gets as an input an arithmetic circuit with the promise that the polyno ..."
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sided error) and evaluates the circuit over the ring of matrices. In addition, we present query complexity lower bounds for identity testing and explore the possibility of derandomizing our algorithm. The analysis of our algorithm uses a noncommutative variant of the SchwartzZippel test. Minimizing algebraic
An argument for Hamiltonicity
, 2008
"... A constantround interactive argument is introduced to show existence of a Hamiltonian cycle in a directed graph. Graph is represented with a characteristic polynomial, top coefficient of a verification polynomial is tested to fit the cycle, soundness follows from SchwartzZippel lemma. 1 ..."
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A constantround interactive argument is introduced to show existence of a Hamiltonian cycle in a directed graph. Graph is represented with a characteristic polynomial, top coefficient of a verification polynomial is tested to fit the cycle, soundness follows from SchwartzZippel lemma. 1
Reducing Randomness Via Irrational Numbers
 In Proceedings of the TwentyNinth Annual ACM Symposium on Theory of Computing
, 1997
"... . We propose a general methodology for testing whether a given polynomial with integer coefficients is identically zero. The methodology evaluates the polynomial at efficiently computable approximations of suitable irrational points. In contrast to the classical technique of DeMillo, Lipton, Schwart ..."
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, Schwartz, and Zippel, this methodology can decrease the error probability by increasing the precision of the approximations instead of using more random bits. Consequently, randomized algorithms that use the classical technique can generally be improved using the new methodology. To demonstrate
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"... Disclaimer: These notes have not been subjected to the usual scrutiny reserved for formal publications.They may be distributed outside this class only with the permission of the Instructor. 2.1 Testing Polynomial Identities Randomized algorithms can be dramatically more efficient than their best kno ..."
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). We will make use of this fact to design an efficientrandomized algorithm for the problem. 2.1.1 The SchwartzZippel Algorithm This algorithm, due independently to Schwartz [S79] and Zippel [Z79], is a Monte Carlo algorithm with abounded probability of false positives and no false negatives