Probabilistic algorithms for sparse polynomials [chapter]

Richard Zippel
<span title="">1979</span> <i title="Springer Berlin Heidelberg"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/2w3awgokqne6te4nvlofavy5a4" style="color: black;">Lecture Notes in Computer Science</a> </i> &nbsp;
Modular approaches [1] to algebraic algorithms, such as tlle GCD, }lave been very useful in cases where there arc very few variables. These algorithms have, unfortunately, an exponential worst ease behaviSr since they need as many as (d + 1) v independent evaluations for a problem with v variables of degree d in each variable. The Hensel lemma was successfully used in many GCD and factorization problems [3, 4, 7, 9, i0] when the problems were sparse. The Hcnsel lemma approach exhibi~,s
more &raquo; ... al behavior at "bad zero" cases that correspond to a zero derivative in the Newton's method analogue. This occurs when substituting zero for one or more variables destroys too much information and reduces the corresponding .lacobian to zero. In such cases it is common to make a linear substitution, such as Y + 3 for Y, in order to avoid tile bad point, The substi~,ution, however, causes a large growth in the size of the revised problem. Thus the Hensel lemma based algorithms tend to run out of space relatively early on bad zero problems. This paper discusses a probabilitistie technique for avoiding the exponential behavior of the modular and Hensel algorithms. This technique's expected running time is a polynomial in the number of terms. Since the results for GCD and factorization can be checked by division, one is guaranteed to obtain the correct answer, if need be, by performing the calculation twice. The probability of getting incorrect results can be made so low, however, that no such backtracking has been required in any of our tests so far. As expected, the experimental results of the algorithms verify the fact that it is exponentially faster than any of the existing algorithms in their worst eases, and its performance is a polynomial function of the size of the final answer in all cases. The probabilistic algorithm presen[ed here will be a variation of the modular GCD algorit, hm. In [ii], we present a formulation of Hensel's lemma that is somewhat more general than the one in current use and our probabilistic analogue to it. Here, we shall only present the modular algorithm. In [11] we shall also discuss how our ideas can be used in computing determinants, resultants and solutions of both linear and non-linear equations. Except in the latter ease, it is relatively diffmult to check the answers, so a smalI probability of error is possible. But as our analysis shows, that probability can be made as low as one pleases, tn [11] , we also discuss the use of our main idea in solving the "intermediate expression swell" problem in those cases where the form of the final answer is known in advance.
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