On 3-matrix factorization of polynomials

We propose a method for the factorization of algebraic polynomials with real or complex coefficients and construct a numerical algorithm, which, along with the factorization of a polynomial with multiple roots, solves the problem of the determination of multiplicities and the number of multiple roots of the polynomial.

By investigation of locus equations we sometimes encounter problems with factorization of resulting polynomials. Commands on factorization of polynomials over the field of rational numbers are implemented in most mathematical software usually by the command factor. We can also use commands on factorization of polynomials over some extension of the field of rational numbers, for instance command AFactor in Maple. Factorization over real or complex numbers is much more difficult. In two examples we will show how to make factorization using dynamic geometry systems in such cases when related commands fail.

Kaltofen (Randomness in computation, vol 5, pp 375---412, 1989) proved the remarkable fact that multivariate polynomial factorization can be done efficiently, in randomized polynomial time. Still, more than twenty years after Kaltofen's work, many questions remain unanswered regarding the complexity aspects of polynomial factorization, such as the question of whether factors of polynomials efficiently computed by arithmetic formulas also have small arithmetic formulas, asked in Kopparty et al. (2014), and the question of bounding the depth of the circuits computing the factors of a polynomial. We are able to answer these questions in the affirmative for the interesting class of polynomials of bounded individual degrees, which contains polynomials such as the determinant and the permanent. We show that if $${P(x_{1},\ldots,x_{n})}$$P(x1,?,xn) is a polynomial with individual degrees bounded by r that can be computed by a formula of size s and depth d, then any factor $${f(x_{1},\ldots, x_{n})}$$f(x1,?,xn) of $${P(x_{1},\ldots,x_{n})}$$P(x1,?,xn) can be computed by a formula of size $${\textsf{poly}((rn)^{r},s)}$$poly((rn)r,s) and depth d + 5. This partially answers the question above posed in Kopparty et al. (2014), who asked if this result holds without the dependence on r. Our work generalizes the main factorization theorem from Dvir et al. (SIAM J Comput 39(4):1279---1293, 2009), who proved it for the special case when the factors are of the form $${f(x_{1}, \ldots, x_{n}) \equiv x_{n} - g(x_{1}, \ldots, x_{n-1})}$$f(x1,?,xn)?xn-g(x1,?,xn-1). Along the way, we introduce several new technical ideas that could be of independent interest when studying arithmetic circuits (or formulas).

Massively parallel search for linear factors in polynomials with many variables

ABSTRACTGuàrdia, Montes and Nart generalized the well-known method of Ore to find complete factorization of polynomials with coefficients in finite extensions of p-adic numbers using Newton polygons of higher order (cf. [Trans. Amer. Math. Soc. 364 (2012), 361–416]). In this paper, we develop the theory of higher order Newton polygons for polynomials with coefficients in henselian valued fields of arbitrary rank and use it to obtain factorization of such polynomials. Our approach is different from the one followed by Guàrdia et al. Some preliminary results needed for proving the main results are also obtained which are of independent interest.

Several standard problems in symbolic computation, such as greatest common divisor and factorization of polynomials, sparse interpolation, or computing solutions to overdetermined systems of polynomial equations have non-trivial solutions only if the input coefficients satisfy certain algebraic constraints. Errors in the coefficients due to floating point round-off or through phsical measurement thus render the exact symbolic algorithms unusable. By symbolic-numeric methods one computes minimal deformations of the coefficients that yield non-trivial results. We will present hybrid algorithms and benchmark computations based on Gauss-Newton optimization, singular value decomposition(SVD) and structure-preserving total least squares (STLS) fitting for several of the above problems.A significant body of results to solve those "approximate computer algebra" problems has been discovered in the past 10 years. In the Computer Algebra Handbook the section on "Hybrid Methods" concludes as follows [2]: "The challenge of hybrid symbolic-numeric algorithms is to explore the effects of imprecision, discontinuity, and algorithmic complexity by applying mathematical optimization, perturbation theory, and inexact arithmetic and other tools in order to solve mathematical problems that today are not solvable by numerical or symbolic methods alone." The focus of our tutorial is on how to formulate several approximate symbolic computation problems as numerical problems in linear algebra and optimization and on software that realizes their solutions.Approximate Greatest Common Divisors [3]. Our paper at this conference presents a solution to the approximate GCD problem for several multivariate polynomials with real or complex coefficients. In addition, the coefficients of the minimally deformed input coefficients can be linearly constrained. In our tutorial we will give a precise definition of the approximate polynomial GCD problem and we will present techniques based on parametric optimization (slow) and STLS or Gauss/Newton iteration (fast) for its numerical solution. The fast methods can compute globally optimal solutions, but they cannot verify global optimality. We show how to apply the constrained approximate GCD problem to computing the nearest singular polynomial with a root of multiplicity at least k≥2.Approximate Factorization of Multivariate Polynomials [1]. Our solution and implementation of the approximate factorization problem follows our approach for the approximate GCD problem. Our algorithms are based on a generalization of the differential forms introduced by W. Ruppert and S. Gao to many variables, and use SVD or STLS and Gauss/Newton optimization to numerically compute the approximate multivariate factors.Solutions of Zero-dimensional Polynomial Systems [4]. We translate a system of polynomials into a system of linear partial differential equations (PDEs) with constant coefficients. The PDEs are brought to an involutive form by symbolic prolongations and numeric projections via SVD. The solutions of the polynomial system are obtained by solving an eigen-problem constructed from the null spaces of the involutive system and its geometric projections.

The Kalman–Yakubovich–Popov Lemma (also called the Yakubovich–Kalman–Popov Lemma) is considered to be one of the cornerstones of Control and Systems Theory due to its applications in absolute stability, hyperstability, dissipativity, passivity, optimal control, adaptive control, stochastic control, and filtering. Despite its broad field of applications, the lemma has been motivated by a very specific problem which is called the absolute stability Lur’e problem [1, 2], and Lur’e’s work in [3] is often quoted as +being the first time the so-called KYP Lemma equations have been introduced. The first results on the Kalman–Yakubovich–Popov Lemma are due to Yakubovich [4, 5] . The proof of Kalman [6] was based on factorization of polynomials, which were very popular among electrical engineers. They later became the starting point for new developments. Using general factorization of matrix polynomials, Popov [7, 8] obtained the lemma in the multivariable case. In the following years, the lemma was further extended to the infinite-dimensional case (Yakubovich [9], Brusin [10], Likhtarnikov and Yakubovich [11]) and discrete-time case (Szego and Kalman [12]).

If f ( x ) f(x) is a polynomial over G F ( q ) GF(q) , we observe (as has Berlekamp) that if h ( x ) q ≡ h ( x ) ( mod f ( x ) ) h{(x)^q} \equiv h(x)(\bmod f(x)) , then f ( x ) = ∏ a ∈ G F ( q ) gcd ( f ( x ) , h ( x ) − a ) f(x) = \prod {{{_a}_{ \in GF(q)}}\gcd (f(x),h(x) - a)} . The object of this paper is to give an explicit construction of enough such h h ’s so that the repeated application of this result will succeed in separating all irreducible factors of f f . The h h ’s chosen are loosely defined by h i ( x ) ≡ x i + x i q + x i q 2 + ⋯ ( mod f ( x ) ) {h_i}(x) \equiv {x^i} + {x^{iq}} + {x^{i{q^2}}} + \cdots (\bmod f(x)) . A detailed example over G F ( 2 ) GF(2) is given, and a table of the factors of the cyclotomic polynomials Φ n ( x ) ( mod p ) for p = 2 , n ≦ 250 ; p = 3 , n ≦ 100 ; p = 5 , 7 , n ≦ 50 {\Phi _n}(x)(\bmod p){\text { for }}p = 2,n \leqq 250;p = 3,n \leqq 100;p = 5,7,n \leqq 50 , is included.

Introduction. A Banach algebra in this paper will be understood to mean a commutative, semi-simple Banach algebra with multiplicative unit e. By the carrier space DA of the Banach algebra A, we mean the space of multiplicative linear functionals on A to C, the complex numbers, and it is to be endowed with the usual weak* topology (cf. [6]). For a E A, a denotes the Gelfand transform of a defined on (DA and A will denote the collection of such functions. We set the following notation. x will be used to denote an indeterminate over A as well as over A and C. If x(x) = 7=Oa,ix' is a polynomial over A, let &(x) and ;,(x) denote, respectively, ,i oQX,(2) = 0}, ac(x) A[x], plays an important role in the present paper. Z(oc(x),A) is topologized with the relative product topology from (A x C. The mapping n is defined by r(h, 2) = h, (h, 2) e Z(oc(x), A). The multiplicity function M of a(x) is defined as follows: for (h, 2) E Z(x(x), A), M(h, A) is equal to the multiplicity of A as a root of cth(x) = 0. In ?1 we introduce the concept of M-neighborhood of a point in Z(a(x),A). We say that W c Z(cx(x), A) is a M-neighborhood of (ho, 2A) E Z(oc(x), A) if W is a neighborhood in Z(oc(x),A) of (ho,AO) and if, for each h e p(W), M(ho, 2O) is equal to the sum of the values of M at the points (h,2) in 7K-1(h) nl W. Proposition 1.1 states that M-neighborhoods exist and that they form a base for the neighborhood system at each point of Z(ox(x),A). The remainder of this section contains most of the topological lemmas needed for our work on factorization. Particular attention is paid to the case where Z(o(x),A) contains a compact open subset K (7t(K) = 'DA) on which M is constant. When this condition obtains, K and 4FD decompose topologically and this decomposition in turn forces a(x) to factor. The main factorization theorem (2.1) says that if a(x)eA[x] and if K (nr(K)-=(A) is a compact open subset of Z(a(x), A), then there exists a monic polynomial P(x) e A[x] such that ,B(x) is a factor of o(x), Z(fl(x), A) = K and Z(ax(x)/fl(x), A) = Z(a(x), A) K. A more detailed description of the factorization of monic polynomials follows. For example, it is shown that if A is indecomposable

Motivated by a connection with the factorization of multivariate polynomials, we study integral convex polytopes and their integral decompositions in the sense of the Minkowski sum. We first show that deciding decomposability of integral polygons is NP-complete then present a pseudo-polynomial-time algorithm for decomposing polygons. For higher-dimensional polytopes, we give a heuristic algorithm which is based upon projections and uses randomization. Applications of our algorithms include absolute irreducibility testing and factorization of polynomials via their Newton polytopes.

In this paper, we develop a new method based on Newton polygon and graded polynomials, similar to the known one based on Newton polygon and residual polynomials. This new method allows us the factorization of any monic polynomial in any henselian valued field. As applications, we give a new proof of Hensel’s lemma and a theorem on prime ideal factorization.

The more sophisticated polynomial algorithms discussed in the following chapters require estimates on the size of the coefficients of factors of a polynomial. The most natural of these results give bounds on the size of linear factors of univariate polynomials,i.e., bounds on the size of zeroes of univariate polynomials. These bounds are very useful when computing the polynomial zeroes numerically. The linear factor estimates can be extended to give bounds on the size of larger degree factors. These estimates are used in calculations of the GCD’s and factorizations of polynomials over the integers.

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We provide trisection (division by 3) algorithms for Jacobians of genus 2 curves over finite fields $$\mathbb {F}_q$$Fq of odd characteristic which rely on the factorization of a polynomial whose roots correspond (bijectively) to the set of trisections of the given divisor. We also construct a polynomial whose roots allow us to calculate the 3-torsion divisors. We show the relation between the rank of the 3-torsion subgroup and the factorization of this 3-torsion polynomial, and describe the factorization of the trisection polynomials in terms of the Galois structure of the 3-torsion subgroup. We also generalize these ideas for $$\ell \in \{5,7\}$$lź{5,7}.

The Kalman–Yakubovich–Popov Lemma (also called the Yakubovich–Kalman–Popov Lemma) is considered to be one of the cornerstones of Control and Systems Theory due to its applications in absolute stability, hyperstability, dissipativity, passivity, optimal control, adaptive control, stochastic control, and filtering. Despite its broad field of applications, the lemma has been motivated by a very specific problem which is called the absolute stability Lur’e problem [1, 2], and Lur’e’s work in [3] is often quoted as +being the first time the so-called KYP Lemma equations have been introduced. The first results on the Kalman–Yakubovich–Popov Lemma are due to Yakubovich [4, 5] . The proof of Kalman [6] was based on factorization of polynomials, which were very popular among electrical engineers. They later became the starting point for new developments. Using general factorization of matrix polynomials, Popov [7, 8] obtained the lemma in the multivariable case. In the following years, the lemma was further extended to the infinite-dimensional case (Yakubovich [9], Brusin [10], Likhtarnikov and Yakubovich [11]) and discrete-time case (Szego and Kalman [12]).