On the degrees of bases of free modulesover a polynomial ring

Given a system of polynomial equations and inequations with coefficients in the field of rational numbers, we show how to compute a geometric resolution of the set of common roots of the system over the field of complex numbers. A geometric resolution consists of a primitive element of the algebraic extension defined by the set of roots, its minimal polynomial, and the parametrizations of the coordinates. Such a representation of the solutions has a long history which goes back to Leopold Kronecker and has been revisited many times in computer algebra. We introduce a new generation of probabilistic algorithms where all the computations use only univariate or bivariate polynomials. We give a new codification of the set of solutions of a positive dimensional algebraic variety relying on a new global version of Newton's iterator. Roughly speaking the complexity of our algorithm is polynomial in some kind of degree of the system, in its height, and linear in the complexity of evaluation of the system. We present our implementation in the Magma system which is called Kronecker in homage to his method for solving systems of polynomial equations. We show that the theoretical complexity of our algorithm is well reflected in practice and we exhibit some cases for which our program is more efficient than the other available software. Academic Press

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“…Even when it is a free module we do not know bases of small size [5]. The next two propositions give some properties of computations in B$.…”

Section: Noether Position Primitive Elementmentioning

confidence: 96%

A Gröbner Free Alternative for Polynomial System Solving

Giusti

Lecerf

Salvy

2001

Journal of Complexity

221

327

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“…In the sequel, we also show a lower bound for ng improving on the one presented in [60], assuming the Generalized Riemann Hypothesis. This lower bound is a consequence of the studies done in [29,28,25,30], Let us observe that if the system S is inconsistent over C, the arithmetic Bezout identity (2) holds. In that case, ng is a bounded function.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 92%

“…In that case, ng is a bounded function. The obvious reason is that if (2) holds, then (3) is false for all those prime numbers p such that a^O (modjj). In particular, if we know the value of a from the arithmetic Nullstellensatz, we immediately obtain the following bound:…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

“…nng is a bounded function. Thus, studies around the arithmetic Bezout identity (2) with estimates for the absolute value of a (and consequently of the coefficients of the polynomials pi,...,ps) are of central relevance for estimating ng.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

“…A discussion of the historical achievements around (arithmetic) Bezout identities follows. Let us assume now that d is an upper bound for the degrees of the polynomials in the input system S = {f\,..., fs} and that h is an upper bound for the absolute values of the coefficients of the polynomials in S. First estimates for log2 |a| were obtained by the systematic use of upper bounds for the degrees of the polynomials pi,...,ps that may occur in a Bezout identity like (2).…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

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On the intrinsic complexity of the arithmetic Nullstellensatz

Hägele

Morais

Pardo

et al. 2000

Journal of Pure and Applied Algebra

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We show several arithmetic estimates for Hilbert's Nullstellensatz. This includes an algorith mic procedure computing the polynomials and constants occurring in a Bezout identity, whose complexity is polynomial in the geometric degree of the system. Moreover, we show for the first time height estimates of intrinsic type for the polynomials and constants appearing, again poly nomial in the geometric degree and linear in the height of the system. These results are based on a suitable representation of polynomials by straight-line programs and duality techniques using the Trace Formula for Gorenstein algebras. As an application we show more precise upper bounds for the function 7ls(x) counting the number of primes yielding an inconsistent modular polynomial equation system. We also give a computationally interesting lower bound for the density of small prime numbers of controlled bit length for the reduction to positive characteris tic of inconsistent systems. Again, this bound is given in terms of intrinsic parameters.

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Computing bases of complete intersection rings in Noether position

Almeida¹,

Blaum²,

D'Alfonso³

et al. 2001

Journal of Pure and Applied Algebra

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Let k be an effective infinite perfect field, k[x1, ..., xn] the polynomial ring in n variables and F ∈ k[x1, ..., xn] M ×M a square polynomial matrix verifying F 2 = F. Suppose that the entries of F are polynomials given by a straight line program of size L and their total degrees are bounded by an integer D. We show that there exists a well parallelizable algorithm which computes bases of the kernel and the image of F in time (nL) O(1) (M D) O(n). By means of this result we obtain a single exponential algorithm to compute a basis of a complete intersection ring in Noether position. More precisely: let f1, ..., fn−r ∈ k[x1, ..., xn] be a regular sequence of polynomials given by a slp of size , whose degrees are bounded by d. Let R := k[x1, ..., xr] and S := k[x1, ..., xn]/ (f1, ..., fn−r) such that S is integral over R; we show that there exists an algorithm running in time O (n) d O(n 2) which computes a basis of S over R. Also, as a consequence of our techniques, we show a single exponential well parallelizable algorithm which decides the freeness of a finite k[x1, ..., xn]−module given by a presentation matrix, and in the affirmative case it computes a basis.

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On the degrees of bases of free modulesover a polynomial ring

Abstract: Let k be an infinite field, A the polynomial ring k[x 1 , .

Cited by 3 publications

References 20 publications

A Gröbner Free Alternative for Polynomial System Solving

A Gröbner Free Alternative for Polynomial System Solving

On the intrinsic complexity of the arithmetic Nullstellensatz

Computing bases of complete intersection rings in Noether position

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