Polar varieties, Bertini's theorems and number of points of singular complete intersections over a finite field

Cafure, Antonio; Matera, Guillermo; Privitelli, Melina

doi:10.1016/j.ffa.2014.09.002

Cited by 9 publications

(14 citation statements)

References 42 publications

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“…This allows us to establish a number of facts concerning the geometry of the set V of solutions of such a polynomial system (see, e.g., Theorems 3.7, 3.11 and 5.1 and Corollary 5.2). Combining these results with estimates on the number of F q -rational points of singular complete intersections of [CMP12a], we obtain our main results (Theorems 4.2 and 5.4).…”

Section: Introductionmentioning

confidence: 55%

“…In what follows, we shall use an estimate on the number of F q -rational points of a projective normal complete intersection of [CMP12a] (see also [CM07] or [GL02] for other estimates). More precisely, if W ⊂ P n is a normal complete intersection defined over F q of dimension n − l ≥ 2, degree δ and multidegree d := (d 1 , .…”

Section: The Number Of Polynomials In a λmentioning

confidence: 99%

“…Combining this with results about the geometry of V of Section 3 we conclude that its projective closure pcl(V ) is regular in codimension 2. Our estimate then follows from estimates on the number of F q -rational points of complete intersections which are regular in codimension 2 due to [CMP12a]. 5.1.…”

Section: The Number Of Polynomials In a λ In The Sparse Casementioning

confidence: 99%

“…5.2. The estimate of |A λ | for large r. In what follows, we shall use an estimate on the number of F q -rational points of a projective singular complete intersection defined over F q due to [CMP12a] (see [GL02] for similar estimates). More precisely, if W ⊂ P n is an F q -definable ideal-theoretic complete intersection of dimension n − l, degree δ ≥ 2, multidegree (d 1 , .…”

Section: The Number Of Polynomials In a λ In The Sparse Casementioning

confidence: 99%

“…. , d l ) and singular locus of dimension at most n − l − 3, then the following estimate holds (see [CMP12a,Corollary 8.4]):…”

Section: The Number Of Polynomials In a λ In The Sparse Casementioning

confidence: 99%

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The distribution of factorization patterns on linear families of polynomials over a finite field

2016

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We obtain estimates on the number |A λ | of elements on a linear family A of monic polynomials of Fq[T ] of degree n having factorization pattern λ := 1 λ 1 2 λ 2 · · · n λn . We show that |A λ | = T (λ) q n−m +O(q n−m−1/2 ), where T (λ) is the proportion of elements of the symmetric group of n elements with cycle pattern λ and m is the codimension of A. Furthermore, if the family A under consideration is "sparse", then |A λ | = T (λ) q n−m + O(q n−m−1 ). Our estimates hold for fields Fq of characteristic greater than 2. We provide explicit upper bounds for the constants underlying the O-notation in terms of λ and A with "good" behavior. Our approach reduces the question to estimate the number of Fq-rational points of certain families of complete intersections defined over Fq. Such complete intersections are defined by polynomials which are invariant under the action of the symmetric group of permutations of the coordinates. This allows us to obtain critical information concerning their singular locus, from which precise estimates on their number of Fq-rational points are established.

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Section: Introductionmentioning

confidence: 55%

Section: The Number Of Polynomials In a λmentioning

confidence: 99%

Section: The Number Of Polynomials In a λ In The Sparse Casementioning

confidence: 99%

Section: The Number Of Polynomials In a λ In The Sparse Casementioning

confidence: 99%

“…. , d l ) and singular locus of dimension at most n − l − 3, then the following estimate holds (see [CMP12a,Corollary 8.4]):…”

Section: The Number Of Polynomials In a λ In The Sparse Casementioning

confidence: 99%

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The distribution of factorization patterns on linear families of polynomials over a finite field

2016

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Factorization patterns on nonlinear families of univariate polynomials over a finite field

2019

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We estimate the number |A λ | of elements on a nonlinear family A of monic polynomials of Fq[T ] of degree r having factorization pattern λ := 1 λ 1 2 λ 2 · · · r λr . Weis the proportion of elements of the symmetric group of r elements with cycle pattern λ and m is the codimension of A. We provide explicit upper bounds for the constants underlying the O-notation in terms of λ and A with "good" behavior. We also apply these results to analyze the average-case complexity of the classical factorization algorithm restricted to A, showing that it behaves as good as in the general case.

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Estimates on the number of Fq–rational solutions of variants of diagonal equations over finite fields

Pérez

Privitelli

2020

Finite Fields and Their Applications

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Polar varieties, Bertini's theorems and number of points of singular complete intersections over a finite field

Cited by 9 publications

References 42 publications

The distribution of factorization patterns on linear families of polynomials over a finite field

The distribution of factorization patterns on linear families of polynomials over a finite field

Factorization patterns on nonlinear families of univariate polynomials over a finite field

Estimates on the number of Fq–rational solutions of variants of diagonal equations over finite fields

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