Zippel R. Probabilistic Algorithms for Sparse Polynomials. Proc. of Eurosam 79', Springer-Verlag LNCS, 72, 216--226, 1979.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....multivariate modular GCD algorithms over Q that one may consider extending to work over L. We have completed an implementation of Brown s algorithm (see [3] which uses rational reconstruction and trial division (see [8] and have begun work on an implementation of Zippel s algorithm (see [11]) We have encountered three bottlenecks on real problems, namely, i) rational reconstruction, ii) the trial divisions, and (iii) extensions of low degree. We will address (i) and (ii) in this paper. Problem (iii) is addressed in [8] We rst give details of the data structure that we use for ....

R. Zippel, Probabilistic algorithms for sparse polynomials, Proceedings of EUROSAM '79, Springer-Verlag LNCS, 2 (1979), pp. 216-226.

....likely preserved [9] with some exceptions [5] Proposition 2.2 Let B be as in Fact 2.1. If diagonal entries in D are chosen uniformly and randomly from a subset of F n f0g with cardinality s then B possesses in addition property c with probability at least 1 2n =s. The Schwartz Zippel Lemma [10, 12] states that if we evaluate a multivariate polynomial of total degree d, with coe cients from F , each variable chosen uniformly and randomly from a subset S of F of size s, then the probability that the result is nonzero is 1 d=s. Proposition 2.2 follows as a corollary of the Schwartz Zippel ....

R. Zippel. Probabilistic algorithms for sparse polynomials. In Proc. EUROSAM 79, pages 216226, Marseille, 1979.

....AN RF OF A MATRIX In this section, we describe how to compute an RF for SPD matrices. We start with some definitions. Fix an SPD n n matrix A, where n is a power of 2, with integer entries of magnitude . 3.1. Random Choice of Modulus Proposition 3.1. See Schwartz [80] and Zippel [84]) Let p be a prime number selected at random from the interval [2(n#A## ) n, 2(n#A## ) for any c 0 2. If det(A) det(A) mod p) with probability 1 #(n log(n#A## ) n#A## ) 3.2. Multipliers for the RF We now define multipliers m# for the matrices A# . Define m = 1 ....

....polynomial. Their algorithm requires 2 log m multiplications of matrices of size at most nmax (n, m) Their reduction is a Las Vegas randomized type of reduction. Using a random, independent choice of the elements of n vector v over a fixed set of polynomial size, the Schwartz Zippel Lemma [84, 80] insures a failure probability . They observe that the matrix powers A, A , A 2 #log m# can be computed in m# stages of matrix products, and that K(A, v) can be computed in log m further stages, where using the identity for i = 1, m#, A v, Av, A v) ....

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R. E. Zippel, Probabilistic algorithms for sparse polynomials, in "Proc. EUROSAM '79", Lecture Notes in Computer Science 72 (1991), 216--226.

.... m univariate factorization and to factoring multivariate polyno ials with coefficients in algebraic number fields and finite fields in polynomial time. 1. Introduction oth the classical Kronecker algorithm [Kronecker r 7 1882] and the modern multivariate Hensel algorithm [Musse 5, Wang 78, Zippel 79] solve the problem of factoring mulf tivariate polynomials with integer coefficients by reduction to actoring univariate polynomials and reconstructing the mul n tivariate factors from the univariate ones. However, the run ing time of both methods suffers from the fact that, in rare f cases, ....

Zippel, R. E.: Probabilistic Algorithms for Sparse Polynomials. Ph.D. thesis, MIT 1979.

....correct and probably fast because one may always check a candidate sparsest shift via any of the variants of the Ben Or Tiwari sparse interpolation algorithm. First, we may choose random values as interpolation points rather than symbolic ones, and employ the probabilistic analysis of [4, 21, 20]. In the univariate case we replace the polynomial root finder by a GCD procedure. This is possible since the sparsest shifts are the roots of a sequence of discrepancies. We can provide a complete probabilistic analysis when the algorithm is run on two independent trials or when all discrepancies ....

Zippel, R. Probabilistic algorithms for sparse polynomials. In Proc. EUROSAM '79 (Heidelberg, 8

....mentioned above, much previous work has also focused on the problem of interpolating a polynomial, i.e. on the problem of exactly identifying a polynomial using membership queries only. As noted above, most of this work has dealt with large fields, such as the real numbers. For instance, Zippel [22, 23], Ben Or and Tiwari [3] and Mansour [18] give efficient algorithms for interpolating sparse multivariate polynomials over such fields. Grigoriev, Karpinski and Singers [13] and Clausen et al. 8] consider the problem of interpolating a sparse polynomial over various finite fields (see also the ....

Richard Zippel. Probabilistic algorithms for sparse polynomials. In Symbolic and Algebraic Computation, pages 216--226. Springer-Verlag, June 1979.

....operations in the element field and our space cost is measured in terms of the number of field elements stored. Finite fields are categorized as large or small, depending on whether they have sufficiently many elements to support these randomized methods for which the Schwartz Zippel lemma [22,26] (see also [3] is used in the probability analysis. We organize solutions around this distinction and offer new results for large and small fields. In Section 2 we offer a list of problems to which preconditioning has been applied with a discussion of the solution methods advanced to date. In ....

R. Zippel, Probabilistic algorithms for sparse polynomials, in: Proc. EUROSAM '79, Heidelberg, Germany, Lecture Notes in Computer Science, vol. 72, Springer, Berlin, 1979, pp. 216--226.

....operations in the element field and our space cost is measured in terms of the number of field elements stored. Finite fields are categorized as large or small, depending on whether they have su#ciently many elements to support those randomized methods for which the Schwartz Zippel lemma [23, 27] (see also [3] is used in the probability analysis. We organize solutions around this distinction and o#er new results for large and small fields. In section 2 we o#er a list of problems to which preconditioning has been applied with a discussion of the solution methods advanced to date. In ....

Zippel, R. Probabilistic algorithms for sparse polynomials. In Proc. EUROSAM '79 (Heidelberg, Germany, 1979), vol. 72 of Lect. Notes Comput. Sci., Springer Verlag, pp. 216--226. 20

....to these two parameters. 1.1 Reducing Field Size The motivation for the following theorem is primarily one of curiosity. The smallest field size over which polynomials of a given total degree exhibit sufficient redundancy to, say, enable the application of the Schwartz Zippel like theorems [16, 18], is when the field size is at least d 2. The low degree tester of [11] uses sets of the same size, i.e. d 2, as elementary test sets. Their proof manages to show that in a certain sense (see Lemma 5) fields of size d 2 are sufficient to show some sort of robustness. However their proof ....

R. Zippel. Probabilistic algorithms for sparse polynomials. EUROSAM '79, Lecture Notes in Computer Science, 72:216--226, 1979.

.... m . Obviously the multivariate polynomial codes form a generalization of the Reed Solomon codes (again using the rst de nition given here of Reed Solomon codes) The distance property of the multivariate polynomial codes follow also from the distance property of multivariate polynomials (cf. [5, 13, 21]) Lemma 6. For integers m; l and q with l q, the code C poly;m;l;q is an [n; k; d] q code with n = q m , k = m l m and d = q l)q m 1 . Proof. The bound on n is immediate. The fact that the number of coecients i 1 ; i m s.t. P j i j l is at m l l is a well known ....

R. E. Zippel. Probabilistic algorithms for sparse polynomials. EUROSAM '79, Lecture Notes in Computer Science, 72:216-226, 1979.

.... Note that the obvious algorithm, namely to tranform the arithmetic expression in a sum of monoms and check whether all coefficients are zero, can have up to exponential running time (in the size of the input) Efficient probabilistic zero tests were developped by Schwartz [Sch80] and Zippel [Zip79] The version below is a variant shown by Ibarra and Moran [IM83] They extended the corresponding theorem for multilinear polynomials shown by Blum, Chandra, and Wegman [BCW80] to arbitrary degrees. Theorem 2.2 [IM83, Sch80, Zip79] Let p(x 1 ; x n ) be a multivariate polynomial of ....

....zero tests were developped by Schwartz [Sch80] and Zippel [Zip79] The version below is a variant shown by Ibarra and Moran [IM83] They extended the corresponding theorem for multilinear polynomials shown by Blum, Chandra, and Wegman [BCW80] to arbitrary degrees. Theorem 2. 2 [IM83, Sch80, Zip79] Let p(x 1 ; x n ) be a multivariate polynomial of degree d over field F that is not the zero polynomial. Let T F with jT j d. Then there are at least (jT j Gamma d) n points (a 1 ; a n ) 2 T n such that p(a 1 ; a n ) 6= 0. We mention two important consequences of ....

R. Zippel. Probabilistic algorithms for sparse polynomials. In ISSAC '79: Proc. Int'l. Symp. on Symbolic and Algebraic Computation, Lecture Notes in Computer Science, Vol. 72. SpringerVerlag, 1979. 20

....wrong Recall that det X = z a1 z b2 z c3 z d4 z a1 z b2 z c4 z d3 z a2 z b1 z c3 z d4 z a2 z b1 z c4 z d3 : The determinant of X is a nonzero polynomial but we get a root of the polynomial by setting each of the variables to 1. Fortunately, as demonstrated by the following theorem of Zippel [18] and Schwartz [16] roots of polynomials are relatively scarce. Theorem 3.1 Let p(x 1 ; x k ) be a nonzero polynomial of degree at most d, and let S be a nite subset of R. If ( x 1 ; x k ) is a random element of S k , then p( x 1 ; x k ) 6= 0 with probability at ....

R.E. Zippel, \Probabilistic algorithms for sparse polynomials", in Proc. EUROSAM 79 (Edward W. Ng, ed.), Lecture Notes in Comput. Sci. 72, Springer{Verlag, Berlin, 1979, 216-226. 24

....at step 0 fails, so success is assured if the change of variables avoids the hypersurface de ned in the last point of Subsection 4.3; this hypersurface has degree at most 4(n 1)d 2n (4d n 1) 2 . We now return to the assumptions made on (p; p 0 ; and use Zippel Schwartz lemma [63, 56] to quantify the choices that assure success. Let be a subset of k, and suppose that the values (p; p 0 ; are chosen in m m m 1 . Besides, we recall that the polynomial has degree at most d n (2d n nd 1) in P 1 ; Pm . There at most d n (2d n nd 1)j j m ....

Zippel, R. Probabilistic algorithms for sparse polynomials. In Proceedings of EUROSAM '79 (1979), no. 72 in Lecture Notes in Computer Science, Springer, pp. 216| 226. 36

....this problem remains open. Nevertheless, there is a min max formula for determining the rank of a mixed skew symmetric matrix; see [10] 2 Randomized algorithms In this section we prove Theorems 1.1 and 1. 2, for which we require the following lemma which was discovered independently by Zippel [22] and Schwartz [19] Lemma 2.1 Let p(x 1 ; x k ) be a nonzero polynomial of degree at most d, and let S be a nite subset of R. If ( x 1 ; x k ) is a random element of S k , then p( x 1 ; x k ) 6= 0 with probability at least 1 d jSj . Proof. The proof is by ....

R.E. Zippel, \Probabilistic algorithms for sparse polynomials", in Proc. EUROSAM 79 (Edward W. Ng, ed.), Lecture Notes in Comput. Sci. 72, Springer{Verlag, Berlin, 1979, 216-226. 12

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Zippel R. Probabilistic Algorithms for Sparse Polynomials. Proc. of Eurosam 79', Springer-Verlag LNCS, 72, 216--226, 1979.

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R. E. Zippel. Probabilistic algorithms for sparse polynomials. In EUROSCAM'79, pages 216-226. Springer LNCS 72, 1979. 15

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R. Zippel, Probabilistic algorithms for sparse polynomials, in: Proc. EUROSAM'79, Marseille 1979.

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R. Zippel, Probabilistic algorithms for sparse polynomials, in: Proceedings EUROSAM' 79, Vol. 72 of Lecture Notes in Computer Science, 1979, pp. 216{ 226. 68

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R. Zippel. Probabilistic algorithms for sparse polynomials. In Proceedings EUROSAM' 79, volume 72 of LNCS, pages 216--226, 1979.

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Richard Zippel. Probabilistic algorithms for sparse polynomials. In Symbolic and algebraic computation (EUROSAM '79, Internat. Sympos., Marseille, 1979), pages 216--226. Springer, Berlin, 1979. 18 A A Fourier lemma In this section we prove Lemma 2 via the Fourier representation of the function. For a boolean

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R. Zippel. Probabilistic algorithms for sparse polynomials. In Proceedings EUROSAM ' 79 , vol. 72 of LNCS , pp. 216-226. 1979.

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Richard Zippel. Probabilistic algorithms for sparse polynomials. In Symbolic and algebraic computation (EUROSAM '79, Internat. Sympos., Marseille, 1979.

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R. Zippel, Probabilistic algorithms for sparse polynomials, Proceedings of EUROSAM '79, Springer-Verlag LNCS, 2 (1979), pp. 216--226.

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R. E. Zippel, Probabilistic algorithms for sparse polynomials, Proceedings of the International Symposium on Symbolic and Algebraic Manipulation (EUROSAM '79) LNCS 72, Springer-Verlag (1979), pp. 216--226.

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R. Zippel, "Probabilistic algorithms for sparse polynomials", Symbolic and algebraic computation (EUROSAM '79, Internat. Sympos., Marseille,

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