Results 1  10
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14
Groebner Basis Under Composition
 J. Symbolic Computation
, 1995
"... this paper is: When does composition commute with Groebner basis computation? We prove that this happens iff the composition is "compatible" with the term ordering and the undivisibility. This has a natural application in computation of Groebner basis of composed polynomials which arise o ..."
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Cited by 14 (3 self)
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this paper is: When does composition commute with Groebner basis computation? We prove that this happens iff the composition is "compatible" with the term ordering and the undivisibility. This has a natural application in computation of Groebner basis of composed polynomials which arise often in reallife problems. 1. Introduction
ON THE EQUATIONS OF THE MOVING CURVE IDEAL OF A RATIONAL ALGEBRAIC PLANE CURVE
, 2009
"... Given a parametrization of a rational plane algebraic curve C, some explicit adjoint pencils on C are described in terms of determinants. Moreover, some generators of the Rees algebra associated to this parametrization are presented. The main ingredient developed in this paper is a detailed study ..."
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Cited by 12 (1 self)
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Given a parametrization of a rational plane algebraic curve C, some explicit adjoint pencils on C are described in terms of determinants. Moreover, some generators of the Rees algebra associated to this parametrization are presented. The main ingredient developed in this paper is a detailed study of the elimination ideal of two homogeneous polynomials in two homogeneous variables that form a regular sequence.
Resultants of composed polynomials
, 2000
"... The objective of this research has been to develop methods for computing resultants of composed polynomials, efficiently, by utilizing their composition structure. By the resultant of several polynomials in several variables (one fewer variables than polynomials) we mean an irreducible polynomial in ..."
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Cited by 10 (7 self)
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The objective of this research has been to develop methods for computing resultants of composed polynomials, efficiently, by utilizing their composition structure. By the resultant of several polynomials in several variables (one fewer variables than polynomials) we mean an irreducible polynomial in the coefficients of the polynomials that vanishes if they have a common zero. By a composed polynomial we mean the polynomial obtained from a given polynomial by replacing each variable by a polynomial. The main motivation for this research comes from the following observations: Resultants of polynomials are frequently computed in many areas of science and in applications because they are fundamentally utilized in solving systems of polynomial equations. Further, polynomials arising in science and applications are often composed because humans tend to structure knowledge modularly and hierarchically. Thus, it is important to have theories and software libraries for efficiently computing resultants of composed polynomials. However, most existing mathematical theories do not adequately support composed polynomials and most algorithms as well as software libraries ignore the composition structure, thus suffering from enormous blow up in space and time. Thus, it is important to develop theories and software libraries for efficiently computing resultants of composed polynomials. The main finding of this research is that resultants of composed polynomials can be nicely factorized, namely, they can be factorized into products of powers of the resultants of the component polynomials and of some of their parts. These factorizations can be utilized to compute resultants of composed polynomials with dramatically improved efficiency.
Double Sylvester sums for subresultants and multiSchur functions
 J. Symbolic Comput
"... Abstract J. J. Sylvester has announced formulas expressing the subresultants (or the successive polynomial remainders for the Euclidean division) of two polynomials, in terms of some double sums over the roots of the two polynomials. We prove Sylvester formulas using the techniques of multivariate ..."
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Cited by 9 (2 self)
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Abstract J. J. Sylvester has announced formulas expressing the subresultants (or the successive polynomial remainders for the Euclidean division) of two polynomials, in terms of some double sums over the roots of the two polynomials. We prove Sylvester formulas using the techniques of multivariate polynomials involving multiSchur functions and divided differences.
Functional Decomposition of Symbolic Polynomials
"... Earlier work has presented algorithms to factor and compute GCDs of symbolic Laurent polynomials, that is multivariate polynomials whose exponents are themselves integervalued polynomials. This article extends the notion of univariate polynomial decomposition to symbolic polynomials and presents an ..."
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Cited by 8 (3 self)
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Earlier work has presented algorithms to factor and compute GCDs of symbolic Laurent polynomials, that is multivariate polynomials whose exponents are themselves integervalued polynomials. This article extends the notion of univariate polynomial decomposition to symbolic polynomials and presents an algorithm to compute these decompositions. For example, the symbolic polynomial f(X) = 2Xn2 +n n − 4X 2 + 2Xn2 −n + 1 can be decomposed as f = g ◦ h where g(X) = 2X2 + 1 and h(X) = Xn2 /2+n/2 n − X 2 /2−n/2. 1.
Resultants of Partially Composed Polynomials
, 2005
"... We study the structure of resultants of two homogeneous partially composed polynomials. By two homogeneous partially composed polynomials we mean a pair of polynomials of which one does not have any given composition structure and the other one is obtained by composing a bivariate homogeneous polyno ..."
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Cited by 5 (4 self)
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We study the structure of resultants of two homogeneous partially composed polynomials. By two homogeneous partially composed polynomials we mean a pair of polynomials of which one does not have any given composition structure and the other one is obtained by composing a bivariate homogeneous polynomial with two bivariate homogeneous polynomials. The main contributions are two equivalent formulas, each representing the resultant of two partially composed polynomials as a certain iterated resultant of the component polynomials. Furthermore, in many cases, this iterated resultant can be computed with dramatically increased efficiency, as demonstrated by experiments.
Groebner Basis Under Composition II
 In Proceedings of ISSAC 96 (International Symposium on Symbolic and Algebraic Computation
, 1996
"... Composition is an operation of replacing variables in a polynomial with other polynomials. The main question of this paper is: When does composition commute with Groebner basis computation (possibly under different term orderings)? We prove that this happens if the leading terms of the composition p ..."
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Cited by 5 (0 self)
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Composition is an operation of replacing variables in a polynomial with other polynomials. The main question of this paper is: When does composition commute with Groebner basis computation (possibly under different term orderings)? We prove that this happens if the leading terms of the composition polynomials form "permuted powering". This is a sequel to another paper where we dealt with a more restricted question (that required same term ordering).
CayleyDixon resultant matrices of multiunivariate composed polynomials
 Computer Algebra in Scientific Computing
, 2005
"... Abstract. The behavior of the CayleyDixon resultant construction and the structure of Dixon matrices are analyzed for composed polynomial systems constructed from a multivariate system in which each variable is substituted by a univariate polynomial in a distinct variable. It is shown that a Dixon ..."
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Cited by 3 (3 self)
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Abstract. The behavior of the CayleyDixon resultant construction and the structure of Dixon matrices are analyzed for composed polynomial systems constructed from a multivariate system in which each variable is substituted by a univariate polynomial in a distinct variable. It is shown that a Dixon projection operator (a multiple of the resultant) of the composed system can be expressed as a power of the resultant of the outer polynomial system multiplied by powers of the leading coefficients of the univariate polynomials substituted for variables in the outer system. The derivation of the resultant formula for the composed system unifies all the known related results in the literature. A new resultant formula is derived for systems where it is known that the CayleyDixon construction does not contain any extraneous factors. The approach demonstrates that the resultant of a composed system can be effectively calculated by considering only the resultant of the outer system. 1
Dresultant and subresultants
 PROC. AMER. MATH. SOC
, 2005
"... We establish a connection between the Dresultant of two polynomials f(t) andg(t) and the subresultant sequence of f(t)−x and g(t)−y. This connection is used to decide in a more explicit way whether K(f(t),g(t)) = K(t) or K[f(t),g(t)] = K[t]. We also show how to extract a faithful parametrizatio ..."
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Cited by 3 (0 self)
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We establish a connection between the Dresultant of two polynomials f(t) andg(t) and the subresultant sequence of f(t)−x and g(t)−y. This connection is used to decide in a more explicit way whether K(f(t),g(t)) = K(t) or K[f(t),g(t)] = K[t]. We also show how to extract a faithful parametrization from a given one.
Multivariate Resultants Under Composition
, 1996
"... this paper is: What happens to multivariate resultants under composition? We prove that the multivariate resultant of composed polynomial is a product of certain powers of the multivariate resultants of the original polynomials and the composition polynomials. This generalizes the well known fact fo ..."
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Cited by 2 (2 self)
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this paper is: What happens to multivariate resultants under composition? We prove that the multivariate resultant of composed polynomial is a product of certain powers of the multivariate resultants of the original polynomials and the composition polynomials. This generalizes the well known fact for the linear case to the nonlinear case. 1. Introduction