Counting reducible and singular bivariate polynomials Joachim von - PowerPoint PPT Presentation

Counting reducible and singular bivariate polynomials Joachim von zur Gathen Bonn 1

Four “accidents” can happen to a bivariate (or multivariate) polynomial over a field: ◮ a nontrivial factor, ◮ a square factor, ◮ a factor over an extension field, ◮ a singular root, where all partial derivatives also vanish. We have a ground field F . The accidents may occur at two places: ◮ in F (“rational”), ◮ in an algebraic closure of F (“absolute”). 2

Four “accidents” can happen to a bivariate (or multivariate) polynomial over a field: ◮ a nontrivial factor, ◮ a square factor, ◮ a factor over an extension field, ◮ a singular root, where all partial derivatives also vanish. We have a ground field F . The accidents may occur at two places: ◮ in F (“rational”), ◮ in an algebraic closure of F (“absolute”). 3

Four “accidents” can happen to a bivariate (or multivariate) polynomial over a field: ◮ a nontrivial factor, ◮ a square factor, ◮ a factor over an extension field, ◮ a singular root, where all partial derivatives also vanish. We have a ground field F . The accidents may occur at two places: ◮ in F (“rational”), ◮ in an algebraic closure of F (“absolute”). 4

Four “accidents” can happen to a bivariate (or multivariate) polynomial over a field: ◮ a nontrivial factor, ◮ a square factor, ◮ a factor over an extension field, ◮ a singular root, where all partial derivatives also vanish. We have a ground field F . The accidents may occur at two places: ◮ in F (“rational”), ◮ in an algebraic closure of F (“absolute”). 5

Overview Introduction Reducible polynomials Squareful polynomials Relatively irreducible polynomials Singular Polynomials 6

Overview Introduction Reducible polynomials Squareful polynomials Relatively irreducible polynomials Singular Polynomials 7

Overview Introduction Reducible polynomials Squareful polynomials Relatively irreducible polynomials Singular Polynomials 8

Overview Introduction Reducible polynomials Squareful polynomials Relatively irreducible polynomials Singular Polynomials 9

Overview Introduction Reducible polynomials Squareful polynomials Relatively irreducible polynomials Singular Polynomials 10

Taxonomy of views on polynomials over finite fields 1 variable 2 variables ≥ 2 variables 11

Taxonomy of views on polynomials over finite fields 1 variable 2 variables ≥ 2 variables 12

Taxonomy of views on polynomials over finite fields 1 variable 2 variables ≥ 2 variables 13

Taxonomy of views on polynomials over finite fields 1 variable 2 variables ≥ 2 variables 14

Taxonomy of views on polynomials over finite fields 1 variable 2 variables ≥ 2 variables total degree degrees in each monic in x 1 variable Gao & Lauder Carlitz, Cohen, Fredman 15

Taxonomy of views on polynomials over finite fields 1 variable 2 variables ≥ 2 variables total degree degrees in each monic in x 1 variable Gao & Lauder Carlitz, Cohen, Fredman 16

Taxonomy of views on polynomials over finite fields 1 variable 2 variables ≥ 2 variables total degree degrees in each monic in x 1 variable Gao & Lauder Carlitz, Cohen, Fredman 17

Taxonomy of views on polynomials over finite fields 1 variable 2 variables ≥ 2 variables total degree degrees in each monic in x 1 variable Gao & Lauder Carlitz, Cohen, Fredman 18

Taxonomy of views on polynomials over finite fields 1 variable 2 variables ≥ 2 variables total degree degrees in each monic in x 1 variable Gao & Lauder Carlitz, Cohen, Fredman 19

Taxonomy of views on polynomials over finite fields 1 variable 2 variables ≥ 2 variables total degree degrees in each monic in x 1 variable Gao & Lauder Carlitz, Cohen, Fredman 20

Taxonomy of views on polynomials over finite fields 1 variable 2 variables ≥ 2 variables total degree degrees in each monic in x 1 variable Gao & Lauder Carlitz, Cohen, Fredman deg ≤ n deg = n Ragot, Lenstra 21

Taxonomy of views on polynomials over finite fields 1 variable 2 variables ≥ 2 variables total degree degrees in each monic in x 1 variable Gao & Lauder Carlitz, Cohen, Fredman deg ≤ n deg = n Ragot, Lenstra 22

Taxonomy of views on polynomials over finite fields 1 variable 2 variables ≥ 2 variables total degree degrees in each monic in x 1 variable Gao & Lauder Carlitz, Cohen, Fredman deg ≤ n deg = n Ragot, Lenstra 23

Taxonomy of views on polynomials over finite fields 1 variable 2 variables ≥ 2 variables total degree degrees in each monic in x 1 variable Gao & Lauder Carlitz, Cohen, Fredman deg ≤ n deg = n Ragot, Lenstra 24

Taxonomy of views on polynomials over finite fields 1 variable 2 variables ≥ 2 variables total degree degrees in each monic in x 1 variable Gao & Lauder Carlitz, Cohen, Fredman deg ≤ n deg = n Ragot, Lenstra exact counting approximate Carlitz, Wan, counting vzG & Viola 25

Taxonomy of views on polynomials over finite fields 1 variable 2 variables ≥ 2 variables total degree degrees in each monic in x 1 variable Gao & Lauder Carlitz, Cohen, Fredman deg ≤ n deg = n Ragot, Lenstra exact counting approximate Carlitz, Wan, counting vzG & Viola 26

Taxonomy of views on polynomials over finite fields 1 variable 2 variables ≥ 2 variables total degree degrees in each monic in x 1 variable Gao & Lauder Carlitz, Cohen, Fredman deg ≤ n deg = n Ragot, Lenstra exact counting approximate Carlitz, Wan, counting vzG & Viola 27

Taxonomy of views on polynomials over finite fields 1 variable 2 variables ≥ 2 variables total degree degrees in each monic in x 1 variable Gao & Lauder Carlitz, Cohen, Fredman deg ≤ n deg = n Ragot, Lenstra exact counting approximate Carlitz, Wan, counting vzG & Viola 28

Taxonomy of views on polynomials over finite fields 1 variable 2 variables ≥ 2 variables total degree degrees in each monic in x 1 variable Gao & Lauder Carlitz, Cohen, Fredman deg ≤ n deg = n Ragot, Lenstra exact counting approximate Carlitz, Wan, counting vzG & Viola error q −O (1) error like q − n 29

Taxonomy of views on polynomials over finite fields 1 variable 2 variables ≥ 2 variables total degree degrees in each monic in x 1 variable Gao & Lauder Carlitz, Cohen, Fredman deg ≤ n deg = n Ragot, Lenstra exact counting approximate Carlitz, Wan, counting vzG & Viola error q −O (1) error like q − n 30

Taxonomy of views on polynomials over finite fields 1 variable 2 variables ≥ 2 variables total degree degrees in each monic in x 1 variable Gao & Lauder Carlitz, Cohen, Fredman deg ≤ n deg = n Ragot, Lenstra exact counting approximate Carlitz, Wan, counting vzG & Viola error q −O (1) error like q − n 31

Notation: ◮ B n ( F ) ⊆ F [ x , y ]: bivariate polynomials with total degree ≤ n . ◮ Certain natural sets A n ( F ) ⊆ B n ( F ). Two different languages: geometric and combinatorial. ◮ Geometry: B n ( F ) affine space over F , A n ( F ) union of images of polynomial maps, thus (reducible) subvariety. Geometric goal: determine the codimension of A n ( F ) = codimension of irreducible components of maximal dimension. ◮ Combinatorial goal: F = F q for a prime power q , find functions α n ( q ) and β n ( q ) so that � � # A n ( F q ) � � # B n ( F q ) − α n ( q ) � ≤ α n ( q ) · β n ( q ) , � � � with β n ( q ) tending to zero as q and n grow. 32

Notation: ◮ B n ( F ) ⊆ F [ x , y ]: bivariate polynomials with total degree ≤ n . ◮ Certain natural sets A n ( F ) ⊆ B n ( F ). Two different languages: geometric and combinatorial. ◮ Geometry: B n ( F ) affine space over F , A n ( F ) union of images of polynomial maps, thus (reducible) subvariety. Geometric goal: determine the codimension of A n ( F ) = codimension of irreducible components of maximal dimension. ◮ Combinatorial goal: F = F q for a prime power q , find functions α n ( q ) and β n ( q ) so that � � # A n ( F q ) � � # B n ( F q ) − α n ( q ) � ≤ α n ( q ) · β n ( q ) , � � � with β n ( q ) tending to zero as q and n grow. 33

Notation: ◮ B n ( F ) ⊆ F [ x , y ]: bivariate polynomials with total degree ≤ n . ◮ Certain natural sets A n ( F ) ⊆ B n ( F ). Two different languages: geometric and combinatorial. ◮ Geometry: B n ( F ) affine space over F , A n ( F ) union of images of polynomial maps, thus (reducible) subvariety. Geometric goal: determine the codimension of A n ( F ) = codimension of irreducible components of maximal dimension. ◮ Combinatorial goal: F = F q for a prime power q , find functions α n ( q ) and β n ( q ) so that � � # A n ( F q ) � � # B n ( F q ) − α n ( q ) � ≤ α n ( q ) · β n ( q ) , � � � with β n ( q ) tending to zero as q and n grow. 34