Kaltofen, E. Effective Noether irreducibility forms and applications. J. Comput. System Sci. 50, 2 (1995), 274--295. 281  Home/Search   Document Not in Database   Summary   Related Articles   Check

....[29, 5, 3, 17] in that it has two parts. First in Section 3. 1 we prove the theorem when m is constant (specifically, m = 2; 3) this uses algebraic arguments inspired by Sudan s [32] work on reconstructing polynomials from very noisy data and Kaltofen s work on Effective Hilbert Irreducibility [20, 21, 22]. Then in Section 3.2 we bootstrap to allow larger m. This part uses probabilistic arguments and relies upon the cases m = 2; 3 (including Theorems 3 and 4 for the cases m = 2; 3) It is inspired by the symmetry based approach of Arora [3] 2 The Low degree Test Let F be a finite field and ....

....This specific substitution has been studied by Kaltofen [21] who bounds the probability with which the polynomial may factor after the substitution, if the substitution is performed randomly . The bound presented in [21] is too weak for our purposes. Fortunately, in a later work Kaltofen [22] presents improved bounds. The bounds in [22] are presented for a different substitution, but the analysis easily extends to the substitution studied in [21] We summarize this theorem below. Theorem 6 ( 20] Let Q 2 F[z; y 1 ; y 2 ; y m ] be a degree l polynomial that is absolutely ....

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E. KALTOFEN. Effective Noether irreducibility forms and applications. Journal of Computer and System Sciences, 50(2):274-295, 1995.

....of L f (Y ) 0. We note that the above method computes the number of irreducible factors in Q(x) Y ] of f in a number of steps that is bounded by a polynomial of the total degree of f and the bit size of the coefficients of f . For more results about absolute factorisation see for example [10] [11] and [6] 3. Factorisation of linear differential equations It is well known that the computation of the Galois group of a polynomial can be reduced to the factorisation of polynomials. A weaker but similar result holds for differential equations [22] In the following we want to consider a ....

E. Kaltofen. Effective Noether irreducibility forms and applications. J. Comput. System Sci., 50(2):274-295, 1995.

.... Recently, polynomial GCD algorithms have been developed that use implicit representations and thus avoid the computationally costly content and primitive part computations needed in those GCD algorithms for polynomials in explicit representation [Kaltofen 1988, Kaltofen and Trager 1990a, D iaz and Kaltofen 1995]. 3.2 Resultants of Multivariate Systems The solvability of a set of nonlinear multivariate polynomials can be determined by the vanishing of a generalization of the Sylvester resultant of two polynomials in a single variable. We examine two generalizations, namely, the classical and the sparse ....

....for this problem. The root is computed as a Taylor series in y, and the integrality of the linear relation for the powers of the series means that the multipliers are polynomials in y of bounded degree. Several algorithms of polynomial time complexity and pointers to the literature are found in [Kaltofen 1995]. Bivariate polynomials constitute implicit representations of algebraic curves. It is an important operation in geometric modeling to convert from implicit to parametric representation. For example, the circle x 2 y 2 Gamma 1 = 0 has the rational parameterization x = 2t 1 t 2 ; y = 1 ....

E. Kaltofen. Effective Noether irreducibility forms and applications. J. Comput. System Sci., 50(2):274--295, 1995.

....and the division Q n i=1 (x n i Gamma 1) Q n i=1 (x i Gamma 1) Both operands have 2 n terms while the quotient has n n terms. Although we consider output sensitive analysis somewhat unnatural, the formulation of our analysis can account for such cases. We follow the approach of [5] and estimate the degrees and coefficients when the algorithm is performed on symbolic inputs. To make our estimates, we use 1 norms of multivariate polynomials over Z, which for a given polynomial are defined as the sum of the absolute values of its integral coefficients. For all f; g 2 Z[y1 ; ....

Kaltofen, E. Effective Noether irreducibility forms and applications. J. Comput. System Sci. 50, 2 (1995), 274--295.

....A94, FS95] in that it has two parts. First in Section 3. 1 we prove the theorem when m is constant (specifically, m = 2; 3) this uses algebraic arguments inspired by Sudan s [Su96] work on reconstructing polynomials from very noisy data and Kaltofen s work on Effective Hilbert Irreducibility [K85, K95]. Then in Section 3.3 we bootstrap to allow larger m. This part uses probabilistic arguments and relies upon the cases m = 2; 3 (including Theorems 6 and 7 for the cases m = 2; 3) It is inspired by the symmetry based approach of Arora [A94] Finally, the appendix contains the construction of ....

E. Kaltofen. Effective Noether irreducibility forms and applications. Journal of Computer and System Sciences, 50(2):274-295, 1995.

....A94, FS95] in that it has two parts. First in Section 3. 1 we prove the theorem when m is constant (specifically, m = 2; 3) this uses algebraic arguments inspired by Sudan s [Su96] work on reconstructing polynomials from very noisy data and Kaltofen s work on Effective Hilbert Irreducibility [K85, K95]. Then in Section 3.3 we bootstrap to allow larger m. This part uses probabilistic arguments and relies upon the cases m = 2; 3 (including Theorems 3 and 4 for the cases m = 2; 3) It is inspired by the symmetry based approach of Arora [A94] Finally, the appendix contains the construction of ....

....on most lines does not have a linear factor. Now we state and prove this fact. It is a simpler version of Kaltofen s Effective Hilbert Irreducibility [K85] in that it focusses only factors that are monic and linear in one of the variables. The proof essentially follows from Kaltofen [K95], and is included here for completeness. A polynomial (in this section, polynomial means a formal polynomial) Q 2 F[z; y 1 ; y m ] is said to be monic with respect to z if the leading coefficient of z is a constant (i.e. an element of F) It is absolutely irreducible if it does not ....

E. Kaltofen. Effective Noether irreducibility forms and applications. Journal of Computer and System Sciences, 50(2):274-295, 1995.

....model and the division i=1 (x i Gamma 1) Q n i=1 (x i Gamma 1) Both operands have terms while the quotient has n terms. Although we consider output sensitive analysis somewhat unnatural, the formulation of our analysis can account for such cases. 278 We follow the approach of [5] and estimate the degrees and coefficients when the algorithm is performed on symbolic inputs. To make our estimates, we use 1 norms of multivariate polynomials over Z, which for a given polynomial are defined as the sum of the absolute values of its integral coefficients. For all f; g 2 Z[y1 ; ....

Kaltofen, E. Effective Noether irreducibility forms and applications. J. Comput. System Sci. 50, 2 (1995), 274--295. 281

....occurring fractions unreduced, as in polynomial GCD computations [Knu81] p. 414, eq. 27) The bit complexity can then be estimated within polynomial bounds because of the lemmas stated in section 2. A similar, but more difficult analysis has been carried out for a generic factorization algorithm [Kal95]. We will also provide some test cases where the input basis contain polynomial entries. Acknowledgment: Imin Chen, then an undergraduate summer research student at Rensselaer Polytechnic Institute visiting from Queen s University at Kingston, Canada and now a Ph.D. student at Oxford University, ....

E. Kaltofen. Effective Noether irreducibility forms and applications. J. Comput. System Sci., 50(2):274--295, 1995.

....for this problem. The root is computed as a Taylor series in y, and the integrality of the linear relation for the powers of the series means that the multipliers are polynomials in y of bounded degree. Several algorithms of polynomial time complexity and pointers to the literature are found in [Kal95] Bivariate polynomials constitute implicit representations of algebraic curves. It is an important operation in geometric modeling to convert from implicit to parametric representation. For example, the circle x 2 y 2 Gamma 1 = 0 has the rational parameterization x = 2t 1 t 2 ; y ....

E. Kaltofen. Effective Noether irreducibility forms and applications. J. Comput. System Sci., 50(2):274--295, 1995.

.... Gamma fk ffl; for a reasonable coefficient vector norm k Delta k: This problem was first posed in my survey article (Kaltofen 1992b) Efficient algorithms for performing the factorization of a multivariate polynomial over the complex 4 E. Kaltofen numbers exactly are described and cited in (Kaltofen 1995). Galligo and Watt (1997) present heuristics for computing complex numerical factors. Since then, I have learned of several related problems and their solution. They are described next. The constrained root problem described below solves Problem 1 if one looks for the nearest polynomial with a ....

....of the linearization of a given problem. Examples are: 1. Petr s Q matrix in Berlekamp s algorithm for factoring polynomials over finite fields (Knuth 1997) Kaltofen and Shoup 1998) 2. resultant matrices for non linear algebraic equations (see Section 5 below) 3. the linear systems in Kaltofen s (1995) algorithm for factoring multivariate polynomials over algebraically closed fields. Linear algebra algorithms for black box matrices are an active subject of research. In Table 1 we list several of them. Giesbrecht s (1997) method for finding integral solutions to sparse linear systems is based on ....

Kaltofen, E. (1995). Effective Noether irreducibility forms and applications. J. Comput. System Sci., 50(2):274--295.

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E. Kaltofen. Effective Noether irreducibility forms and applications. In Proc. 22nd Annual ACM Symp. Theory Comput., pages 54--63, New York, N.Y., 1991. ACM Press. Full version in [85]. KaPa91

....file. The items thus retrieved are copyrighted by the publishers or by E. Kaltofen. 2 Major Research Results 2. 1 Polynomial Factorization ffl Polynomial time algorithms for multivariate polynomial factorization with coefficients from a field [5, 6, 25, 27] or the algebraic closure of a field [19, 85]; deterministic polynomial time irreducibility testing of multivariate polynomials over a large finite field [35] ffl Polynomial time sparse multivariate polynomial factorization algorithms by introducing the straight line program [42, 39, 24, 20, 31] and the black box representations of ....

E. Kaltofen. Effective Noether irreducibility forms and applications. J. Comput. System Sci., 50(2):274--295, 1995. DHKLV95

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