In traditional computer algebra on polynomials, we have assumed that the coefficients of polynomi... more In traditional computer algebra on polynomials, we have assumed that the coefficients of polynomials were given rigorously by integers, rational numbers or algebraic numbers and that manipulation on the polynomials were also exact. However, in many practical applications or real world problems, the coefficients contain errors, that is, polynomials
2019 21st International Symposium on Symbolic and Numeric Algorithms for Scientific Computing (SYNASC), 2019
This paper proposes a new method of enhancing Buchberger's algorithm for the lexicographic or... more This paper proposes a new method of enhancing Buchberger's algorithm for the lexicographic order (LEX-order) Groebner bases. The idea is to append polynomials to the input system, where we generate polynomials to be appended by computing PRSs (polynomial remainder sequences) of input polynomials and making them close to basis elements by the GCD operation. In order to do so, we first restrict the input system to satisfy three reasonable conditions (we call such systems "healthy"), and we compute redundant PRSs so that the variable-eliminated remainders are not in a triangular form but in a rectangular form; we call the corresponding PRSs "rectangular PRSs (rectPRSs)"; see 2. A for details of rectPRSs. The lowest order element of the LEX-order Groebner basis can be computed from the last set of remainders of rectPRSs and the GCD operation; see [20]. We generate other polynomials to be appended by computing rectPRSs of leading coefficients of mutually similar r...
International Symposium on Symbolic and Algebraic Computation, 2007
... below. Following Collins [1], we introduce associated polynomial. Let Pi = ci1T1 + ··· + cilT... more ... below. Following Collins [1], we introduce associated polynomial. Let Pi = ci1T1 + ··· + cilTl (i = 1,...,k) be polynomials with T1,...,Tl monomials, and M = cij be a k×l matrix, k<l. The polynomial associated with M, which we denote by assP(M), is defined as follows. assP ...
Let F (x, u1, . . . , u`), with ` ≥ 2, be an irreducible multivariate polynomial over a number fi... more Let F (x, u1, . . . , u`), with ` ≥ 2, be an irreducible multivariate polynomial over a number field K of characteristic 0. A point (s1, . . . , s`) ∈ K̄ is called a singular point for the Hensel construction, if F (x, s1, . . . , s`) has multiple roots. The generalized Hensel construction at the expansion point (s1, . . . , s`) breaks down for the multiple factors of F (x, s1, . . . , s`), and the extended Hensel construction (EHC in short) is the Hensel construction at a singular point [Kuo89],[SK99],[SI00]. Introducing weights for subvariables as (u1, . . . , u`) 7→ (t1u1, . . . , t`u`), with (w1, . . . , w`) ∈ N, and changing the weights variously, we obtain different sets of extended Hensel factors. Then, three questions arise: Q1) how many different sets of Hensel factors can we get?, Q2) how can we classify all the different sets of Hensel factors?, and Q3) what is the relationship between different sets of Hensel factors? McDonald [McD95] considered these questions in the framework of conventional Newton-Puiseux scheme, but his result is very restrictive. In this poster, we answer to these questions in the framework of EHC. The first step of EHC is to determine the so-called Newton polynomial; mutually prime factors of the Newton polynomial are used as initial Hensel factors. We have obtained three theorems. Theorem 1: if the Newton polynomials are the same for
Regarding the Cabibbo angle and p.-e mass splitting, we propose that the intrinsic mass •of a par... more Regarding the Cabibbo angle and p.-e mass splitting, we propose that the intrinsic mass •of a particle is not characterized by underlying symmetry but is connected closely with participating interactions. Following this idea, we introduce another kind of weak interaction which may provide p.-e mass splitting. It is suggested that the mass accompanied by an interaction provides a finite p.-meson mass. Furthermore, it is possible to produce symmetry breaking of quarks and a non-vanishing Cabibbo angle. Since the discovery of SU(3) symmetry, its breaking has been an important problem in physics. Furthermore, discoveries of the Cabibbo angle 1 l and nonelectromagnetic 1-spin breaking 2 ),*) have complicated the problem. In the following, we shall cosider only mass splitting as symmetry breaking. The conventional approach is basically that symmetry principles or invariance principles hold exactly, were there not for dynamical interplays of strong, weak and electromagnetic interactions (from now on we use the abbreviations H 8 , Hw, and He.m for them). Some of the theories based on that approach are as follows: (i) Symmetrical bootstrap of H 8 • 3 ) (ii) . Spontaneous breakdown based on Lagrangian models. 4 l Both of the above theories are able to explain the breakdown of SU(3) symmetry into SU(2) symmetry, producing the Gell-Mann-Okubo mass formula. They, however, are ineffective for treating the problem of the Cabibbo angle. For example, considerations based on theory (i) give a vanishing Cabibbo angle. This result is a general one when He.m and Hw are treated as perturbations. 6
New expressions for particle distrjbutions due to energy momentum conservation constraint in the ... more New expressions for particle distrjbutions due to energy momentum conservation constraint in the longitudinal phase space with rapidity-ordering are obtained by improving the Chew-Pignotti-DeTa r calculations. Five particles at most are taken into consideration in the calculations. A threshold correction is also taken into account by giving the correct kinematical boundaries. Numerical calculations are performed at 24 GeV /c and 1000 GeV /c within the framework of dominant pion production and a vector-meson-domin ance model. Furthermore, we present a simple and practical expression for phase space volume by modifying the Chew-Pignotti-DeTa r one.
Contrary to that the general Hensel construction (GHC: [3]) uses univariate initial Hensel factor... more Contrary to that the general Hensel construction (GHC: [3]) uses univariate initial Hensel factors, the extended Hensel construction (EHC: [8]) uses multivariate initial Hensel factors determined by the Newton polygon (see below) of the given multivariate polynomial F (x, u ) ∈ K[ x , u ], where ( u ) = ( u 1 ,..., u ℓ ), with ℓ ≥ 2, and K is a number field. The F ( x, u ) may be such that its leading coefficient may vanish at ( u ) = ( 0 ) = (0,...,0), and even may be F ( x , 0 ) = 0. The EHC was used so far for computing series expansion of multivariate algebraic function determined by F ( x, u ) = 0, at critical points [8, 5] and for factorization [4, 1] and GCD computation [7] of F ( x , u ), without shifting the origin of u . It allows us to construct efficient algorithms for sparse multivariate polynomials [1, 7]. The EHC is another and promising approach than Zippel's sparse Hensel lifting [9, 10].
2015 17th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing (SYNASC), 2015
Let F(x, u1,..., uℓ) be a squarefree multivariate polynomial in main variable x and sub-variables... more Let F(x, u1,..., uℓ) be a squarefree multivariate polynomial in main variable x and sub-variables u1,..., uℓ. We say that the leading coefficient (LC) of F is singular if it vanishes at the origin of sub-variables. A representative algorithm for nonsparse multivariate polynomial GCD is the EZ-GCD algorithm, which is based on the generalized Hensel construction (GHC). In order to apply the GHC easily, we require 1) the LC of F is nonsingular, 2) F(x, 0,..., 0) is squarefree, and 3) the initial Hensel factor of GCD is "lucky". These requirements are usually satisfied by the "nonzero substitution", i.e., to shift the origin of subvariables. However, the nonzero substitution may cause a drastic increase of the number of terms of F if F is sparse. In 1993, Sasaki and Kako proposed the extended Hensel construction (EHC) which does not perform the nonzero substitution even if the LC is singular. Using the EHC, Inaba implemented an algorithm of multivariate polynomial factorization and verified that it is very useful for sparse polynomials. In this paper, we apply the EHC for the computation of GCD of sparse multivariate polynomials. In order to find a lucky initial factor, we utilize the weighting of sub-variables, etc. Our naive implementation in Maple shows that our algorithm is comparable in performance to Maple's GCD routine based on the sparse interpolation.
2016 18th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing (SYNASC), 2016
The extended Hensel construction (EHC) is a direct extension of the generalized Hensel constructi... more The extended Hensel construction (EHC) is a direct extension of the generalized Hensel construction (GHC), and it targets sparse multivariate polynomials for which the GHC breaks down. The EHC consists of two Hensel constructions which we call separation of "maximal" and "minimal" Hensel factors (see the text). As for the minimal Hensel factor separation, very recently, we enhanced the old algorithm largely by using Groebner basis of two initial factors and syzygies for the elements of the basis. In this paper, we first improve the old algorithm for maximal Hensel factors. We then enhance further the Groebner basis computation in our recent algorithm. The latter is based on a theoretical analysis of the Groebner bases. Simple experiments show that the improved part for the minimal Hensel factors is much faster than the recent one.
A new model of multiple production developed from the optical model is proposed and is examined i... more A new model of multiple production developed from the optical model is proposed and is examined in the energy region about 50""1200GeV. Numerical calculations are made by treating it as a potential problem and by making use of Glauber's high energy approxima• tion. The result is not unreasonable. In particular the longitudinal momentum distribution does not show the scaling behavior above "'-'1000GeV.
Four new algorithms for multivariate polynomial GCD (greatest common divisor) are given. The firs... more Four new algorithms for multivariate polynomial GCD (greatest common divisor) are given. The first is a simple improvement of PRS (polynomial remainder sequence) algorithms. The second is to calculate a Groebner basis with a certain term ordering. The third is to calculate subresultant by treating the coefficients as truncated power series. The fourth is to calculate PRS by treating the coefficients as truncated power series. The first and second algorithms are not important practically, but the third and fourth ones are quite efficient and seem to be useful practi-cally. 1.
In traditional computer algebra on polynomials, we have assumed that the coefficients of polynomi... more In traditional computer algebra on polynomials, we have assumed that the coefficients of polynomials were given rigorously by integers, rational numbers or algebraic numbers and that manipulation on the polynomials were also exact. However, in many practical applications or real world problems, the coefficients contain errors, that is, polynomials
2019 21st International Symposium on Symbolic and Numeric Algorithms for Scientific Computing (SYNASC), 2019
This paper proposes a new method of enhancing Buchberger's algorithm for the lexicographic or... more This paper proposes a new method of enhancing Buchberger's algorithm for the lexicographic order (LEX-order) Groebner bases. The idea is to append polynomials to the input system, where we generate polynomials to be appended by computing PRSs (polynomial remainder sequences) of input polynomials and making them close to basis elements by the GCD operation. In order to do so, we first restrict the input system to satisfy three reasonable conditions (we call such systems "healthy"), and we compute redundant PRSs so that the variable-eliminated remainders are not in a triangular form but in a rectangular form; we call the corresponding PRSs "rectangular PRSs (rectPRSs)"; see 2. A for details of rectPRSs. The lowest order element of the LEX-order Groebner basis can be computed from the last set of remainders of rectPRSs and the GCD operation; see [20]. We generate other polynomials to be appended by computing rectPRSs of leading coefficients of mutually similar r...
International Symposium on Symbolic and Algebraic Computation, 2007
... below. Following Collins [1], we introduce associated polynomial. Let Pi = ci1T1 + ··· + cilT... more ... below. Following Collins [1], we introduce associated polynomial. Let Pi = ci1T1 + ··· + cilTl (i = 1,...,k) be polynomials with T1,...,Tl monomials, and M = cij be a k×l matrix, k<l. The polynomial associated with M, which we denote by assP(M), is defined as follows. assP ...
Let F (x, u1, . . . , u`), with ` ≥ 2, be an irreducible multivariate polynomial over a number fi... more Let F (x, u1, . . . , u`), with ` ≥ 2, be an irreducible multivariate polynomial over a number field K of characteristic 0. A point (s1, . . . , s`) ∈ K̄ is called a singular point for the Hensel construction, if F (x, s1, . . . , s`) has multiple roots. The generalized Hensel construction at the expansion point (s1, . . . , s`) breaks down for the multiple factors of F (x, s1, . . . , s`), and the extended Hensel construction (EHC in short) is the Hensel construction at a singular point [Kuo89],[SK99],[SI00]. Introducing weights for subvariables as (u1, . . . , u`) 7→ (t1u1, . . . , t`u`), with (w1, . . . , w`) ∈ N, and changing the weights variously, we obtain different sets of extended Hensel factors. Then, three questions arise: Q1) how many different sets of Hensel factors can we get?, Q2) how can we classify all the different sets of Hensel factors?, and Q3) what is the relationship between different sets of Hensel factors? McDonald [McD95] considered these questions in the framework of conventional Newton-Puiseux scheme, but his result is very restrictive. In this poster, we answer to these questions in the framework of EHC. The first step of EHC is to determine the so-called Newton polynomial; mutually prime factors of the Newton polynomial are used as initial Hensel factors. We have obtained three theorems. Theorem 1: if the Newton polynomials are the same for
Regarding the Cabibbo angle and p.-e mass splitting, we propose that the intrinsic mass •of a par... more Regarding the Cabibbo angle and p.-e mass splitting, we propose that the intrinsic mass •of a particle is not characterized by underlying symmetry but is connected closely with participating interactions. Following this idea, we introduce another kind of weak interaction which may provide p.-e mass splitting. It is suggested that the mass accompanied by an interaction provides a finite p.-meson mass. Furthermore, it is possible to produce symmetry breaking of quarks and a non-vanishing Cabibbo angle. Since the discovery of SU(3) symmetry, its breaking has been an important problem in physics. Furthermore, discoveries of the Cabibbo angle 1 l and nonelectromagnetic 1-spin breaking 2 ),*) have complicated the problem. In the following, we shall cosider only mass splitting as symmetry breaking. The conventional approach is basically that symmetry principles or invariance principles hold exactly, were there not for dynamical interplays of strong, weak and electromagnetic interactions (from now on we use the abbreviations H 8 , Hw, and He.m for them). Some of the theories based on that approach are as follows: (i) Symmetrical bootstrap of H 8 • 3 ) (ii) . Spontaneous breakdown based on Lagrangian models. 4 l Both of the above theories are able to explain the breakdown of SU(3) symmetry into SU(2) symmetry, producing the Gell-Mann-Okubo mass formula. They, however, are ineffective for treating the problem of the Cabibbo angle. For example, considerations based on theory (i) give a vanishing Cabibbo angle. This result is a general one when He.m and Hw are treated as perturbations. 6
New expressions for particle distrjbutions due to energy momentum conservation constraint in the ... more New expressions for particle distrjbutions due to energy momentum conservation constraint in the longitudinal phase space with rapidity-ordering are obtained by improving the Chew-Pignotti-DeTa r calculations. Five particles at most are taken into consideration in the calculations. A threshold correction is also taken into account by giving the correct kinematical boundaries. Numerical calculations are performed at 24 GeV /c and 1000 GeV /c within the framework of dominant pion production and a vector-meson-domin ance model. Furthermore, we present a simple and practical expression for phase space volume by modifying the Chew-Pignotti-DeTa r one.
Contrary to that the general Hensel construction (GHC: [3]) uses univariate initial Hensel factor... more Contrary to that the general Hensel construction (GHC: [3]) uses univariate initial Hensel factors, the extended Hensel construction (EHC: [8]) uses multivariate initial Hensel factors determined by the Newton polygon (see below) of the given multivariate polynomial F (x, u ) ∈ K[ x , u ], where ( u ) = ( u 1 ,..., u ℓ ), with ℓ ≥ 2, and K is a number field. The F ( x, u ) may be such that its leading coefficient may vanish at ( u ) = ( 0 ) = (0,...,0), and even may be F ( x , 0 ) = 0. The EHC was used so far for computing series expansion of multivariate algebraic function determined by F ( x, u ) = 0, at critical points [8, 5] and for factorization [4, 1] and GCD computation [7] of F ( x , u ), without shifting the origin of u . It allows us to construct efficient algorithms for sparse multivariate polynomials [1, 7]. The EHC is another and promising approach than Zippel's sparse Hensel lifting [9, 10].
2015 17th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing (SYNASC), 2015
Let F(x, u1,..., uℓ) be a squarefree multivariate polynomial in main variable x and sub-variables... more Let F(x, u1,..., uℓ) be a squarefree multivariate polynomial in main variable x and sub-variables u1,..., uℓ. We say that the leading coefficient (LC) of F is singular if it vanishes at the origin of sub-variables. A representative algorithm for nonsparse multivariate polynomial GCD is the EZ-GCD algorithm, which is based on the generalized Hensel construction (GHC). In order to apply the GHC easily, we require 1) the LC of F is nonsingular, 2) F(x, 0,..., 0) is squarefree, and 3) the initial Hensel factor of GCD is "lucky". These requirements are usually satisfied by the "nonzero substitution", i.e., to shift the origin of subvariables. However, the nonzero substitution may cause a drastic increase of the number of terms of F if F is sparse. In 1993, Sasaki and Kako proposed the extended Hensel construction (EHC) which does not perform the nonzero substitution even if the LC is singular. Using the EHC, Inaba implemented an algorithm of multivariate polynomial factorization and verified that it is very useful for sparse polynomials. In this paper, we apply the EHC for the computation of GCD of sparse multivariate polynomials. In order to find a lucky initial factor, we utilize the weighting of sub-variables, etc. Our naive implementation in Maple shows that our algorithm is comparable in performance to Maple's GCD routine based on the sparse interpolation.
2016 18th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing (SYNASC), 2016
The extended Hensel construction (EHC) is a direct extension of the generalized Hensel constructi... more The extended Hensel construction (EHC) is a direct extension of the generalized Hensel construction (GHC), and it targets sparse multivariate polynomials for which the GHC breaks down. The EHC consists of two Hensel constructions which we call separation of "maximal" and "minimal" Hensel factors (see the text). As for the minimal Hensel factor separation, very recently, we enhanced the old algorithm largely by using Groebner basis of two initial factors and syzygies for the elements of the basis. In this paper, we first improve the old algorithm for maximal Hensel factors. We then enhance further the Groebner basis computation in our recent algorithm. The latter is based on a theoretical analysis of the Groebner bases. Simple experiments show that the improved part for the minimal Hensel factors is much faster than the recent one.
A new model of multiple production developed from the optical model is proposed and is examined i... more A new model of multiple production developed from the optical model is proposed and is examined in the energy region about 50""1200GeV. Numerical calculations are made by treating it as a potential problem and by making use of Glauber's high energy approxima• tion. The result is not unreasonable. In particular the longitudinal momentum distribution does not show the scaling behavior above "'-'1000GeV.
Four new algorithms for multivariate polynomial GCD (greatest common divisor) are given. The firs... more Four new algorithms for multivariate polynomial GCD (greatest common divisor) are given. The first is a simple improvement of PRS (polynomial remainder sequence) algorithms. The second is to calculate a Groebner basis with a certain term ordering. The third is to calculate subresultant by treating the coefficients as truncated power series. The fourth is to calculate PRS by treating the coefficients as truncated power series. The first and second algorithms are not important practically, but the third and fourth ones are quite efficient and seem to be useful practi-cally. 1.
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Papers by Tateaki Sasaki