The number of reducible space curves over a finite field

Cesaratto, Eda; Gathen, Joachim von zur; Matera, Guillermo

doi:10.1016/j.jnt.2012.08.027

Cited by 6 publications

(9 citation statements)

References 21 publications

(45 reference statements)

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“…Furthermore, for a finite field we provide nearly optimal bounds on the number of polynomial sequences that define such degenerate varieties. This result generalizes the corresponding one of Cesaratto et al (2013) from curves to projective varieties of arbitrary dimension.…”

Section: Introductionsupporting

confidence: 78%

“…It is shown that its largest irreducible component consists of planar irreducible curves provided that δ is large enough. Over a finite field, Cesaratto et al (2013) use this to obtain estimates, close to 1, on the probability that a uniformly random curve defined over a finite field F q is absolutely irreducible and planar. The present paper shows that for a fixed Bézout number, a typical sequence of polynomials with corresponding degree pattern defines an irreducible hypersurface V in some linear projective subspace of P n K (Theorem 5.7).…”

Section: Introductionmentioning

confidence: 99%

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Explicit estimates for polynomial systems defining irreducible smooth complete intersections

Gathen

Matera

2019

Acta Arith.

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This paper deals with properties of the algebraic variety defined as the set of zeros of a "typical" sequence of polynomials. We consider various types of "nice" varieties: set-theoretic and ideal-theoretic complete intersections, absolutely irreducible ones, and nonsingular ones. For these types, we present a nonzero "obstruction" polynomial of explicitly bounded degree in the coefficients of the sequence that vanishes if its variety is not of the type. Over finite fields, this yields bounds on the number of such sequences. We also show that most sequences (of at least two polynomials) define a degenerate variety, namely an absolutely irreducible nonsingular hypersurface in some linear projective subspace.

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Section: Introductionsupporting

confidence: 78%

Section: Introductionmentioning

confidence: 99%

Explicit estimates for polynomial systems defining irreducible smooth complete intersections

Gathen

Matera

2019

Acta Arith.

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“…The results of Sections 2.1-2.5 are from von zur Gathen, Viola & Ziegler (2013) unless otherwise attributed, those of Section 2.6 are from Cesaratto, von zur Gathen & Matera (2013), and those of Section 2.7 are from von zur Gathen (2011).…”

Section: Counting Multivariate Polynomialsmentioning

confidence: 99%

“…Can we say something similar for other types of varieties? Cesaratto, von zur Gathen & Matera (2013) give an affirmative answer for curves in P r for arbitrary r. A first question is how to parametrize the curves. Moduli spaces only include irreducible curves, and systems of defining equations do not work except for complete intersections.…”

Section: Introductionmentioning

confidence: 99%

Survey on Counting Special Types of Polynomials

Gathen¹,

Ziegler²

2015

Lecture Notes in Computer Science

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Most integers are composite and most univariate polynomials over a finite field are reducible. The Prime Number Theorem and a classical result of Gauß count the remaining ones, approximately and exactly.For polynomials in two or more variables, the situation changes dramatically. Most multivariate polynomials are irreducible. This survey presents counting results for some special classes of multivariate polynomials over a finite field, namely the the reducible ones, the s-powerful ones (divisible by the s-th power of a nonconstant polynomial), the relatively irreducible ones (irreducible but reducible over an extension field), the decomposable ones, and also for reducible space curves. These come as exact formulas and as approximations with relative errors that essentially decrease exponentially in the input size.Furthermore, a univariate polynomial f is decomposable if f = g • h for some nonlinear polynomials g and h. It is intuitively clear that the decomposable polynomials form a small minority among all polynomials. The tame case, where the characteristic p of F q does not divide n = deg f , is fairly well-understood, and we obtain closely matching upper and lower bounds on the number of decomposable polynomials. In the wild case, where p does divide n, the bounds are less satisfactory, in particular when p is the smallest prime divisor of n and divides n exactly twice. The crux of the matter is to count the number of collisions, where essentially different (g, h) yield the same f . We present a classification of all collisions at degree n = p 2 which yields an exact count of those decomposable polynomials.

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“…Can we say something similar for other types of varieties? Cesaratto et al (2013) give an affirmative answer for curves in P r for arbitrary r. A first question is how to parametrize the curves. Moduli spaces only include irreducible curves, and systems of defining equations do not work except for complete intersections.…”

Section: Introductionmentioning

confidence: 99%

Computer Algebra and Polynomials

2015

Lecture Notes in Computer Science

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The number of reducible space curves over a finite field

Cited by 6 publications

References 21 publications

Explicit estimates for polynomial systems defining irreducible smooth complete intersections

Explicit estimates for polynomial systems defining irreducible smooth complete intersections

Survey on Counting Special Types of Polynomials

Computer Algebra and Polynomials

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