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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

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Counting reducible and singular bivariate polynomials

Joachim von zur Gathen Bonn

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Four “accidents” can happen to a bivariate (or multivariate) polynomial

  • ver 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

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Four “accidents” can happen to a bivariate (or multivariate) polynomial

  • ver 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

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Four “accidents” can happen to a bivariate (or multivariate) polynomial

  • ver 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

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Four “accidents” can happen to a bivariate (or multivariate) polynomial

  • ver 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

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Overview

Introduction Reducible polynomials Squareful polynomials Relatively irreducible polynomials Singular Polynomials

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Overview

Introduction Reducible polynomials Squareful polynomials Relatively irreducible polynomials Singular Polynomials

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Overview

Introduction Reducible polynomials Squareful polynomials Relatively irreducible polynomials Singular Polynomials

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Overview

Introduction Reducible polynomials Squareful polynomials Relatively irreducible polynomials Singular Polynomials

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Overview

Introduction Reducible polynomials Squareful polynomials Relatively irreducible polynomials Singular Polynomials

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Taxonomy of views on polynomials over finite fields

1 variable 2 variables ≥ 2 variables

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Taxonomy of views on polynomials over finite fields

1 variable 2 variables ≥ 2 variables

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Taxonomy of views on polynomials over finite fields

1 variable 2 variables ≥ 2 variables

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Taxonomy of views on polynomials over finite fields

1 variable 2 variables ≥ 2 variables

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Taxonomy of views on polynomials over finite fields

1 variable 2 variables ≥ 2 variables total degree degrees in each variable Carlitz, Cohen, Fredman monic in x1 Gao & Lauder

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Taxonomy of views on polynomials over finite fields

1 variable 2 variables ≥ 2 variables total degree degrees in each variable Carlitz, Cohen, Fredman monic in x1 Gao & Lauder

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Taxonomy of views on polynomials over finite fields

1 variable 2 variables ≥ 2 variables total degree degrees in each variable Carlitz, Cohen, Fredman monic in x1 Gao & Lauder

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Taxonomy of views on polynomials over finite fields

1 variable 2 variables ≥ 2 variables total degree degrees in each variable Carlitz, Cohen, Fredman monic in x1 Gao & Lauder

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Taxonomy of views on polynomials over finite fields

1 variable 2 variables ≥ 2 variables total degree degrees in each variable Carlitz, Cohen, Fredman monic in x1 Gao & Lauder

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Taxonomy of views on polynomials over finite fields

1 variable 2 variables ≥ 2 variables total degree degrees in each variable Carlitz, Cohen, Fredman monic in x1 Gao & Lauder

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Taxonomy of views on polynomials over finite fields

1 variable 2 variables ≥ 2 variables total degree degrees in each variable Carlitz, Cohen, Fredman monic in x1 Gao & Lauder deg ≤ n Ragot, Lenstra deg = n

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Taxonomy of views on polynomials over finite fields

1 variable 2 variables ≥ 2 variables total degree degrees in each variable Carlitz, Cohen, Fredman monic in x1 Gao & Lauder deg ≤ n Ragot, Lenstra deg = n

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Taxonomy of views on polynomials over finite fields

1 variable 2 variables ≥ 2 variables total degree degrees in each variable Carlitz, Cohen, Fredman monic in x1 Gao & Lauder deg ≤ n Ragot, Lenstra deg = n

23

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Taxonomy of views on polynomials over finite fields

1 variable 2 variables ≥ 2 variables total degree degrees in each variable Carlitz, Cohen, Fredman monic in x1 Gao & Lauder deg ≤ n Ragot, Lenstra deg = n

24

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Taxonomy of views on polynomials over finite fields

1 variable 2 variables ≥ 2 variables total degree degrees in each variable Carlitz, Cohen, Fredman monic in x1 Gao & Lauder deg ≤ n Ragot, Lenstra deg = n exact counting Carlitz, Wan, vzG & Viola approximate counting

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Taxonomy of views on polynomials over finite fields

1 variable 2 variables ≥ 2 variables total degree degrees in each variable Carlitz, Cohen, Fredman monic in x1 Gao & Lauder deg ≤ n Ragot, Lenstra deg = n exact counting Carlitz, Wan, vzG & Viola approximate counting

26

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Taxonomy of views on polynomials over finite fields

1 variable 2 variables ≥ 2 variables total degree degrees in each variable Carlitz, Cohen, Fredman monic in x1 Gao & Lauder deg ≤ n Ragot, Lenstra deg = n exact counting Carlitz, Wan, vzG & Viola approximate counting

27

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Taxonomy of views on polynomials over finite fields

1 variable 2 variables ≥ 2 variables total degree degrees in each variable Carlitz, Cohen, Fredman monic in x1 Gao & Lauder deg ≤ n Ragot, Lenstra deg = n exact counting Carlitz, Wan, vzG & Viola approximate counting

28

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Taxonomy of views on polynomials over finite fields

1 variable 2 variables ≥ 2 variables total degree degrees in each variable Carlitz, Cohen, Fredman monic in x1 Gao & Lauder deg ≤ n Ragot, Lenstra deg = n exact counting Carlitz, Wan, vzG & Viola approximate counting error q−O(1) error like q−n

29

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Taxonomy of views on polynomials over finite fields

1 variable 2 variables ≥ 2 variables total degree degrees in each variable Carlitz, Cohen, Fredman monic in x1 Gao & Lauder deg ≤ n Ragot, Lenstra deg = n exact counting Carlitz, Wan, vzG & Viola approximate counting error q−O(1) error like q−n

30

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Taxonomy of views on polynomials over finite fields

1 variable 2 variables ≥ 2 variables total degree degrees in each variable Carlitz, Cohen, Fredman monic in x1 Gao & Lauder deg ≤ n Ragot, Lenstra deg = n exact counting Carlitz, Wan, vzG & Viola approximate counting error q−O(1) error like q−n

31

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Notation:

◮ Bn(F) ⊆ F[x, y]: bivariate polynomials with total degree ≤ n. ◮ Certain natural sets An(F) ⊆ Bn(F).

Two different languages: geometric and combinatorial.

◮ Geometry: Bn(F) affine space over F, An(F) union of images of

polynomial maps, thus (reducible) subvariety. Geometric goal: determine the codimension of An(F) = codimension of irreducible components of maximal dimension.

◮ Combinatorial goal: F = Fq for a prime power q, find functions

αn(q) and βn(q) so that

  • #An(Fq)

#Bn(Fq) − αn(q)

  • ≤ αn(q) · βn(q),

with βn(q) tending to zero as q and n grow.

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Notation:

◮ Bn(F) ⊆ F[x, y]: bivariate polynomials with total degree ≤ n. ◮ Certain natural sets An(F) ⊆ Bn(F).

Two different languages: geometric and combinatorial.

◮ Geometry: Bn(F) affine space over F, An(F) union of images of

polynomial maps, thus (reducible) subvariety. Geometric goal: determine the codimension of An(F) = codimension of irreducible components of maximal dimension.

◮ Combinatorial goal: F = Fq for a prime power q, find functions

αn(q) and βn(q) so that

  • #An(Fq)

#Bn(Fq) − αn(q)

  • ≤ αn(q) · βn(q),

with βn(q) tending to zero as q and n grow.

33

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Notation:

◮ Bn(F) ⊆ F[x, y]: bivariate polynomials with total degree ≤ n. ◮ Certain natural sets An(F) ⊆ Bn(F).

Two different languages: geometric and combinatorial.

◮ Geometry: Bn(F) affine space over F, An(F) union of images of

polynomial maps, thus (reducible) subvariety. Geometric goal: determine the codimension of An(F) = codimension of irreducible components of maximal dimension.

◮ Combinatorial goal: F = Fq for a prime power q, find functions

αn(q) and βn(q) so that

  • #An(Fq)

#Bn(Fq) − αn(q)

  • ≤ αn(q) · βn(q),

with βn(q) tending to zero as q and n grow.

34

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Notation:

◮ Bn(F) ⊆ F[x, y]: bivariate polynomials with total degree ≤ n. ◮ Certain natural sets An(F) ⊆ Bn(F).

Two different languages: geometric and combinatorial.

◮ Geometry: Bn(F) affine space over F, An(F) union of images of

polynomial maps, thus (reducible) subvariety. Geometric goal: determine the codimension of An(F) = codimension of irreducible components of maximal dimension.

◮ Combinatorial goal: F = Fq for a prime power q, find functions

αn(q) and βn(q) so that

  • #An(Fq)

#Bn(Fq) − αn(q)

  • ≤ αn(q) · βn(q),

with βn(q) tending to zero as q and n grow.

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Thus a random element of Bn(Fq) is in An(Fq) with probability about αn(q).

◮ Best results: βn(q) goes to zero like q−n. ◮ Simpler results: αn(q) = q−m with βn(q) = O(q−1). ◮ Weil bounds: βn(q)= nO(1)q−1/2.

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Thus a random element of Bn(Fq) is in An(Fq) with probability about αn(q).

◮ Best results: βn(q) goes to zero like q−n. ◮ Simpler results: αn(q) = q−m with βn(q) = O(q−1). ◮ Weil bounds: βn(q)= nO(1)q−1/2.

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Thus a random element of Bn(Fq) is in An(Fq) with probability about αn(q).

◮ Best results: βn(q) goes to zero like q−n. ◮ Simpler results: αn(q) = q−m with βn(q) = O(q−1). ◮ Weil bounds: βn(q)= nO(1)q−1/2.

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Thus a random element of Bn(Fq) is in An(Fq) with probability about αn(q).

◮ Best results: βn(q) goes to zero like q−n. ◮ Simpler results: αn(q) = q−m with βn(q) = O(q−1). ◮ Weil bounds: βn(q)= nO(1)q−1/2.

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Fq[x, y]

singular rationally singular absolutely singular reducible absolutely reducible rationally reducible squareful 1 q ≤ 1 q3/2 1 qn−1 ǫ 1 q2n−1 40

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Reducible polynomials

Fq[x, y]

singular rationally singular absolutely singular reducible absolutely reducible rationally reducible squareful 41

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n all reducibles 1 q3 − q 2 q6 − q3 (q5 + q4 − q2 − q)/2 3 q10 − q6 (3q8 + 2q7 − 2q6 − 3q5 − q4 + 2q3 − q)/3 4 q15 − q10 (4q12 + 6q11 − 2q10 − 5q9 − 7q8 + 6q6 − 2q4 − q3 +q2)/4 5 q21 − q15 (5q17 + 5q16 + 5q15 − 10q13 − 15q12 − 6q11 +11q10 + 10q9 − 5q7 − q6 + q5 + q3 − q)/5 6 q28 − q21 (6q23 + 6q22 + 6q20 + 3q19 − 3q18 − 21q17 −23q16 − 10q15 + 18q14 + 32q13 + 10q12 − 15q11 −12q10 + 3q8 − q7 + 2q5 − 3q3 + q2 + q)/6 The numbers of reducible polynomials of degrees up to 6

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Theorem

Consider polynomials of degree n ≥ 2.

  • 1. {reducibles} is a subvariety of codimension n − 1 in {all}.
  • 2. Let ρn(q) = (q + 1)q−n. Then for n ≥ 3
  • #{reducibles}

#{all} − ρn(q)

  • ≤ ρn(q) · 2q−n+3,

at degree 2: #{reducibles} #{all} = ρ2(q) 2 .

  • 3. For n ≥ 6, we have
  • #{reducibles}

#{all} − q−n+1

  • ≤ 2q−n.

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For 1 ≤ k < n: multiplication map µk,n : {degree k} × {degree n − k} − → {degree n}, (g, h) − → g · h, {reducibles} =

  • 1≤k≤n/2

im µk,n. Multiplication by units gives fiber dimension ≥ 1 = ⇒ Zariski closure of im µk,n is a proper irreducible subvariety = ⇒ complement (= irreducible polynomials) is dense. g, h irreducible = ⇒ fiber dimension = 1. = ⇒ generic fiber dimension is 1, bn = dim{polynomials of degree n}. dim im µk,n = bk + bn−k − 1 = bn − k(n − k). Maximum at k = 1: bn − n + 1. Hence codim = n − 1.

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For 1 ≤ k < n: multiplication map µk,n : {degree k} × {degree n − k} − → {degree n}, (g, h) − → g · h, {reducibles} =

  • 1≤k≤n/2

im µk,n. Multiplication by units gives fiber dimension ≥ 1 = ⇒ Zariski closure of im µk,n is a proper irreducible subvariety = ⇒ complement (= irreducible polynomials) is dense. g, h irreducible = ⇒ fiber dimension = 1. = ⇒ generic fiber dimension is 1, bn = dim{polynomials of degree n}. dim im µk,n = bk + bn−k − 1 = bn − k(n − k). Maximum at k = 1: bn − n + 1. Hence codim = n − 1.

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For 1 ≤ k < n: multiplication map µk,n : {degree k} × {degree n − k} − → {degree n}, (g, h) − → g · h, {reducibles} =

  • 1≤k≤n/2

im µk,n. Multiplication by units gives fiber dimension ≥ 1 = ⇒ Zariski closure of im µk,n is a proper irreducible subvariety = ⇒ complement (= irreducible polynomials) is dense. g, h irreducible = ⇒ fiber dimension = 1. = ⇒ generic fiber dimension is 1, bn = dim{polynomials of degree n}. dim im µk,n = bk + bn−k − 1 = bn − k(n − k). Maximum at k = 1: bn − n + 1. Hence codim = n − 1.

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For 1 ≤ k < n: multiplication map µk,n : {degree k} × {degree n − k} − → {degree n}, (g, h) − → g · h, {reducibles} =

  • 1≤k≤n/2

im µk,n. Multiplication by units gives fiber dimension ≥ 1 = ⇒ Zariski closure of im µk,n is a proper irreducible subvariety = ⇒ complement (= irreducible polynomials) is dense. g, h irreducible = ⇒ fiber dimension = 1. = ⇒ generic fiber dimension is 1, bn = dim{polynomials of degree n}. dim im µk,n = bk + bn−k − 1 = bn − k(n − k). Maximum at k = 1: bn − n + 1. Hence codim = n − 1.

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For 1 ≤ k < n: multiplication map µk,n : {degree k} × {degree n − k} − → {degree n}, (g, h) − → g · h, {reducibles} =

  • 1≤k≤n/2

im µk,n. Multiplication by units gives fiber dimension ≥ 1 = ⇒ Zariski closure of im µk,n is a proper irreducible subvariety = ⇒ complement (= irreducible polynomials) is dense. g, h irreducible = ⇒ fiber dimension = 1. = ⇒ generic fiber dimension is 1, bn = dim{polynomials of degree n}. dim im µk,n = bk + bn−k − 1 = bn − k(n − k). Maximum at k = 1: bn − n + 1. Hence codim = n − 1.

48

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For 1 ≤ k < n: multiplication map µk,n : {degree k} × {degree n − k} − → {degree n}, (g, h) − → g · h, {reducibles} =

  • 1≤k≤n/2

im µk,n. Multiplication by units gives fiber dimension ≥ 1 = ⇒ Zariski closure of im µk,n is a proper irreducible subvariety = ⇒ complement (= irreducible polynomials) is dense. g, h irreducible = ⇒ fiber dimension = 1. = ⇒ generic fiber dimension is 1, bn = dim{polynomials of degree n}. dim im µk,n = bk + bn−k − 1 = bn − k(n − k). Maximum at k = 1: bn − n + 1. Hence codim = n − 1.

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Let n ≥ 3. Each fiber of µk,n has at least q − 1 elements. # im µk,n ≤ 1 q − 1 · #{degree k} · #{degree n − k} < qbk(1 − q−k−1) · qbn−k q − 1 = ρn(q) · {all} · qn−1−k(n−k)(1 − q−k−1) (1 − q−2)(1 − q−n−1) .

◮ Some calculation gives the upper bound for q ≥ 3. ◮ More calculation for q = 2 and n ≥ 8. ◮ Even more for q = 2 and n = 6. ◮ One more for q = 2 and n = 6.

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Let n ≥ 3. Each fiber of µk,n has at least q − 1 elements. # im µk,n ≤ 1 q − 1 · #{degree k} · #{degree n − k} < qbk(1 − q−k−1) · qbn−k q − 1 = ρn(q) · {all} · qn−1−k(n−k)(1 − q−k−1) (1 − q−2)(1 − q−n−1) .

◮ Some calculation gives the upper bound for q ≥ 3. ◮ More calculation for q = 2 and n ≥ 8. ◮ Even more for q = 2 and n = 6. ◮ One more for q = 2 and n = 6.

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Let n ≥ 3. Each fiber of µk,n has at least q − 1 elements. # im µk,n ≤ 1 q − 1 · #{degree k} · #{degree n − k} < qbk(1 − q−k−1) · qbn−k q − 1 = ρn(q) · {all} · qn−1−k(n−k)(1 − q−k−1) (1 − q−2)(1 − q−n−1) .

◮ Some calculation gives the upper bound for q ≥ 3. ◮ More calculation for q = 2 and n ≥ 8. ◮ Even more for q = 2 and n = 6. ◮ One more for q = 2 and n = 6.

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Let n ≥ 3. Each fiber of µk,n has at least q − 1 elements. # im µk,n ≤ 1 q − 1 · #{degree k} · #{degree n − k} < qbk(1 − q−k−1) · qbn−k q − 1 = ρn(q) · {all} · qn−1−k(n−k)(1 − q−k−1) (1 − q−2)(1 − q−n−1) .

◮ Some calculation gives the upper bound for q ≥ 3. ◮ More calculation for q = 2 and n ≥ 8. ◮ Even more for q = 2 and n = 6. ◮ One more for q = 2 and n = 6.

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Let n ≥ 3. Each fiber of µk,n has at least q − 1 elements. # im µk,n ≤ 1 q − 1 · #{degree k} · #{degree n − k} < qbk(1 − q−k−1) · qbn−k q − 1 = ρn(q) · {all} · qn−1−k(n−k)(1 − q−k−1) (1 − q−2)(1 − q−n−1) .

◮ Some calculation gives the upper bound for q ≥ 3. ◮ More calculation for q = 2 and n ≥ 8. ◮ Even more for q = 2 and n = 6. ◮ One more for q = 2 and n = 6.

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Corollary

We have for n ≥ 2 #{irreducibles} ≥ qbn · (1 − (q + 2)q−n). Lower bound: g, h irreducible, k < n/2, = ⇒ fiber size is q − 1, = ⇒ lower bound on reducibles.

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Corollary

We have for n ≥ 2 #{irreducibles} ≥ qbn · (1 − (q + 2)q−n). Lower bound: g, h irreducible, k < n/2, = ⇒ fiber size is q − 1, = ⇒ lower bound on reducibles.

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Corollary

We have for n ≥ 2 #{irreducibles} ≥ qbn · (1 − (q + 2)q−n). Lower bound: g, h irreducible, k < n/2, = ⇒ fiber size is q − 1, = ⇒ lower bound on reducibles.

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Corollary

We have for n ≥ 2 #{irreducibles} ≥ qbn · (1 − (q + 2)q−n). Lower bound: g, h irreducible, k < n/2, = ⇒ fiber size is q − 1, = ⇒ lower bound on reducibles.

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Previous work:

◮ Carlitz 1963:

fraction of irreducibles − 1 = O((q − 1)q−n−1). “As the referee pointed out, [it] can be proved by a crude counting argument” that 1 − q−n+4 (q − 1)3 ≤ fraction of irreducibles ≤ 1.

◮ Carlitz 1965, Cohen 1968, 1970: fraction of irreducibles is

1 − q−m + O(nq−(m+n+1)) among polynomials of degrees m ≤ n in x, y, respectively.

◮ Corresponding results for multivariate polynomials.

59

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Previous work:

◮ Carlitz 1963:

fraction of irreducibles − 1 = O((q − 1)q−n−1). “As the referee pointed out, [it] can be proved by a crude counting argument” that 1 − q−n+4 (q − 1)3 ≤ fraction of irreducibles ≤ 1.

◮ Carlitz 1965, Cohen 1968, 1970: fraction of irreducibles is

1 − q−m + O(nq−(m+n+1)) among polynomials of degrees m ≤ n in x, y, respectively.

◮ Corresponding results for multivariate polynomials.

60

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Previous work:

◮ Carlitz 1963:

fraction of irreducibles − 1 = O((q − 1)q−n−1). “As the referee pointed out, [it] can be proved by a crude counting argument” that 1 − q−n+4 (q − 1)3 ≤ fraction of irreducibles ≤ 1.

◮ Carlitz 1965, Cohen 1968, 1970: fraction of irreducibles is

1 − q−m + O(nq−(m+n+1)) among polynomials of degrees m ≤ n in x, y, respectively.

◮ Corresponding results for multivariate polynomials.

61

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Previous work:

◮ Carlitz 1963:

fraction of irreducibles − 1 = O((q − 1)q−n−1). “As the referee pointed out, [it] can be proved by a crude counting argument” that 1 − q−n+4 (q − 1)3 ≤ fraction of irreducibles ≤ 1.

◮ Carlitz 1965, Cohen 1968, 1970: fraction of irreducibles is

1 − q−m + O(nq−(m+n+1)) among polynomials of degrees m ≤ n in x, y, respectively.

◮ Corresponding results for multivariate polynomials.

62

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Previous work:

◮ Carlitz 1963:

fraction of irreducibles − 1 = O((q − 1)q−n−1). “As the referee pointed out, [it] can be proved by a crude counting argument” that 1 − q−n+4 (q − 1)3 ≤ fraction of irreducibles ≤ 1.

◮ Carlitz 1965, Cohen 1968, 1970: fraction of irreducibles is

1 − q−m + O(nq−(m+n+1)) among polynomials of degrees m ≤ n in x, y, respectively.

◮ Corresponding results for multivariate polynomials.

63

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◮ Cohen 1968 comes to “a fairly long, complicated argument, which

we shall omit”, and warns the interested reader that “the derivation

  • f the above results is increasingly complicated. Each further

computation, using this method, would require considerable calculation.”

◮ Ragot 1997 shows:

q−n+1(1 − 5 q ) ≤ fraction of reducibles ≤ q−n+1(1 + 6 q ).

◮ Gao & Lauder 2002, for polynomials monic in x. ◮ Bodin 2007: relative error bound of 1 n for large enough n.

64

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◮ Cohen 1968 comes to “a fairly long, complicated argument, which

we shall omit”, and warns the interested reader that “the derivation

  • f the above results is increasingly complicated. Each further

computation, using this method, would require considerable calculation.”

◮ Ragot 1997 shows:

q−n+1(1 − 5 q ) ≤ fraction of reducibles ≤ q−n+1(1 + 6 q ).

◮ Gao & Lauder 2002, for polynomials monic in x. ◮ Bodin 2007: relative error bound of 1 n for large enough n.

65

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◮ Cohen 1968 comes to “a fairly long, complicated argument, which

we shall omit”, and warns the interested reader that “the derivation

  • f the above results is increasingly complicated. Each further

computation, using this method, would require considerable calculation.”

◮ Ragot 1997 shows:

q−n+1(1 − 5 q ) ≤ fraction of reducibles ≤ q−n+1(1 + 6 q ).

◮ Gao & Lauder 2002, for polynomials monic in x. ◮ Bodin 2007: relative error bound of 1 n for large enough n.

66

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“Self-reducibility”: Upper bound on reducibles = ⇒ lower bound on irreducibles = ⇒ lower bound on reducibles, by induction

67

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“Self-reducibility”: Upper bound on reducibles = ⇒ lower bound on irreducibles = ⇒ lower bound on reducibles, by induction

68

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“Self-reducibility”: Upper bound on reducibles = ⇒ lower bound on irreducibles = ⇒ lower bound on reducibles, by induction

69

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“Self-reducibility”: Upper bound on reducibles = ⇒ lower bound on irreducibles = ⇒ lower bound on reducibles, by induction

70

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Squareful polynomials

Fq[x, y]

singular rationally singular absolutely singular reducible absolutely reducible rationally reducible squareful 71

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n squareful polynomials 1 2 q3 − q 3 q5 + q4 − q3 − q2 4 q8 + q7 + q6 − 2q5 − 2q4 + q2 5 q12 + q11 − q7 − 2q6 − q5 + q4 + q3 6 q17 + q16 − q12 + q10 − q9 − 4q8 − q7 + 2q6 + 3q5 − q3 The number of squareful polynomials of degrees up to 6.

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Theorem

Let n ≥ 1.

  • 1. For n ≥ 2, {squareful} is a subvariety of codimension 2n − 1.
  • 2. Let

ηn(q) = (q + 1)q−2n(1 − q−n+1) 1 − q−n−1 . Then |fraction of squareful − ηn(q)| ≤ ηn(q) · 3q−2n+6, and for n ≤ 3 fraction of squareful = ηn(q). Cohen 1970: fraction of r-power-free polynomials is 1 − q−rm + O(q−nm) among polynomials of degrees at most m ≤ n in x, y, respectively.

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Theorem

Let n ≥ 1.

  • 1. For n ≥ 2, {squareful} is a subvariety of codimension 2n − 1.
  • 2. Let

ηn(q) = (q + 1)q−2n(1 − q−n+1) 1 − q−n−1 . Then |fraction of squareful − ηn(q)| ≤ ηn(q) · 3q−2n+6, and for n ≤ 3 fraction of squareful = ηn(q). Cohen 1970: fraction of r-power-free polynomials is 1 − q−rm + O(q−nm) among polynomials of degrees at most m ≤ n in x, y, respectively.

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Relatively irreducible polynomials

Fq[x, y]

singular rationally singular absolutely singular reducible absolutely reducible rationally reducible squareful 75

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An irreducible bivariate polynomial is relatively irreducible if it is not absolutely irreducible. Then it is the product of all conjugates of an irreducible polynomial over some extension field. Application: algorithms for curves: point finding, estimating the size. Huang & Ierardi, 1993; von zur Gathen, Shparlinski & Karpinski, 1993, 1996; von zur Gathen & Shparlinski 1995, 1998; Matera & Cafure 2006.

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An irreducible bivariate polynomial is relatively irreducible if it is not absolutely irreducible. Then it is the product of all conjugates of an irreducible polynomial over some extension field. Application: algorithms for curves: point finding, estimating the size. Huang & Ierardi, 1993; von zur Gathen, Shparlinski & Karpinski, 1993, 1996; von zur Gathen & Shparlinski 1995, 1998; Matera & Cafure 2006.

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n relatively irreducibles 2 (q5 − q4 − q2 + q)/2 3 (q7 − q6 + q4 − 2q3 + q)/3 4 (2q11 − 2q10 + q9 − q8 − 2q6 + 2q4 + q3 − q2)/4 5 (q11 − q10 + q6 − q5 − q3 + q)/5 6 (3q19 − 3q18 + 3q17 − q16 − 2q15 − 2q13 + 2q12 −3q11 + 3q8 + q7 − 2q5 + 3q3 − q2 − q)/6 The numbers of relatively irreducible polynomials of degrees up to 6.

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Theorem

Let n ≥ 2, let l ≥ 2 be the smallest prime divisor of n, εn(q) = q−n2(l−1)/2l(1 − q−1) l(1 − q−l)(1 − q−n−1), δn(q) = 2q−2n+2 if n is prime, 2q−n+l+1

  • therwise.

Then 1.

  • fraction of rel irred − εn(q)
  • ≤ εn(q) · δn(q).
  • 2. εn(q) ≤ q−n2/4/2.
  • 3. If n is prime, then εn(q) ≤ q−n(n−1)/2/n and

#{rel irred} = (q − 1)(q2n + qn − q2 − q)/n.

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Theorem

Let n ≥ 2, let l ≥ 2 be the smallest prime divisor of n, εn(q) = q−n2(l−1)/2l(1 − q−1) l(1 − q−l)(1 − q−n−1), δn(q) = 2q−2n+2 if n is prime, 2q−n+l+1

  • therwise.

Then 1.

  • fraction of rel irred − εn(q)
  • ≤ εn(q) · δn(q).
  • 2. εn(q) ≤ q−n2/4/2.
  • 3. If n is prime, then εn(q) ≤ q−n(n−1)/2/n and

#{rel irred} = (q − 1)(q2n + qn − q2 − q)/n.

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Theorem

Let n ≥ 2, let l ≥ 2 be the smallest prime divisor of n, εn(q) = q−n2(l−1)/2l(1 − q−1) l(1 − q−l)(1 − q−n−1), δn(q) = 2q−2n+2 if n is prime, 2q−n+l+1

  • therwise.

Then 1.

  • fraction of rel irred − εn(q)
  • ≤ εn(q) · δn(q).
  • 2. εn(q) ≤ q−n2/4/2.
  • 3. If n is prime, then εn(q) ≤ q−n(n−1)/2/n and

#{rel irred} = (q − 1)(q2n + qn − q2 − q)/n.

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Theorem

Let n ≥ 2, let l ≥ 2 be the smallest prime divisor of n, εn(q) = q−n2(l−1)/2l(1 − q−1) l(1 − q−l)(1 − q−n−1), δn(q) = 2q−2n+2 if n is prime, 2q−n+l+1

  • therwise.

Then 1.

  • fraction of rel irred − εn(q)
  • ≤ εn(q) · δn(q).
  • 2. εn(q) ≤ q−n2/4/2.
  • 3. If n is prime, then εn(q) ≤ q−n(n−1)/2/n and

#{rel irred} = (q − 1)(q2n + qn − q2 − q)/n.

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Singular polynomials

f ∈ F[x, y], P = (u, v) ∈ F 2 : f (P) = 0 ⇐ ⇒ P is on the curve V (f ) ⊆ F 2 ⇐ ⇒ f ∈ mp = (x − u, y − v) ⊆ F[x, y] maximal ideal. f (P) = ∂f ∂x (P) = ∂f ∂y (P) = 0 ⇐ ⇒ P is singular on V (f ) ⇐ ⇒ f is singular at P ⇐ ⇒ f ∈ sp = m2

p.

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Quotient ring F[x, y]/sP =F + (x − u)F + (y − v)F is a 3-dimensional vector space over F. codimF[x,y]sP = 3.

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Affine Hilbert function of sP: codim sP = 3 at degree n for n large enough. Ragot 1997, 1999: fraction of singular = 1 − (1 − q−3)q2 (1) for n > 4q − 2. Similar result for multivariate polynomials.

Theorem

(Lenstra 2006): (1) ⇐ ⇒ n ≥ 3q − 2.

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Affine Hilbert function of sP: codim sP = 3 at degree n for n large enough. Ragot 1997, 1999: fraction of singular = 1 − (1 − q−3)q2 (1) for n > 4q − 2. Similar result for multivariate polynomials.

Theorem

(Lenstra 2006): (1) ⇐ ⇒ n ≥ 3q − 2.

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Affine Hilbert function of sP: codim sP = 3 at degree n for n large enough. Ragot 1997, 1999: fraction of singular = 1 − (1 − q−3)q2 (1) for n > 4q − 2. Similar result for multivariate polynomials.

Theorem

(Lenstra 2006): (1) ⇐ ⇒ n ≥ 3q − 2.

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Affine Hilbert function of sP: codim sP = 3 at degree n for n large enough. Ragot 1997, 1999: fraction of singular = 1 − (1 − q−3)q2 (1) for n > 4q − 2. Similar result for multivariate polynomials.

Theorem

(Lenstra 2006): (1) ⇐ ⇒ n ≥ 3q − 2.

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R = Fq[x, y]: P ∈ F2

q, random polynomial:

prob(singular at P) = q−3 prob(nonsingular at P) = 1 − q−3

  • P∈F2

q

R/sP, random polynomial: × · · · × prob(nonsingular at all P) = (1 − q−3)q2 Independence: Chinese Remainder Theorem

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R = Fq[x, y]: P ∈ F2

q, random polynomial:

prob(singular at P) = q−3 prob(nonsingular at P) = 1 − q−3

  • P∈F2

q

R/sP, random polynomial: × · · · × prob(nonsingular at all P) = (1 − q−3)q2 Independence: Chinese Remainder Theorem

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  • P∈F2

q

R/mP =

  • u,v∈Fq

R/(x − u, y − v) = R/

  • u,v∈Fq

(x − u, y − v) = R/(xq − x, y q − y) Monomial xiy j ↔ (i, j):

  • · · ·
  • .

. . . . . ... . . .

  • · · ·
  • · · ·
  • q − 1

q − 1 j i (q − 1, q − 1) Representatives for R/(xq − x, y q − y)

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Representation of

  • P∈F2

q

R/sP =

  • P∈F2

q

R/m2

P

= R/(

  • P∈F2

q

m2

P) = R/(xq − x, y q − y)2

= R/((xq − x)2, (xq − x)(y q − y), (y q − y)2).

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  • ...
  • .

. . ...

  • · · ·
  • · · ·
  • .

. . ... . . . . . . ...

  • · · ·
  • · · ·
  • ...
  • q − 1

q 2q − 1 (0, n) q − 1q 2q − 1 (n, 0) j i

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(1) holds ⇔ degree n → R/(xq − x, y q − y)2 surjective ⇔ n ≥ 3q − 2.

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Small n? 1 − (1 − q−3)q2 = q2 1

  • q−3 −

q2 2

  • q−6 + − · · ·

≈ q−1 − 1 2q−2 + − · · ·

Theorem

  • 1. {singular} is an irreducible subvariety with codimension 1.
  • 2. For q, n ≥ 3, we have

q−1 − 1 2q−2 ≤ fraction of singular ≤ q−1.

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Theorem

The fraction τ of absolutely singular and rationally nonsingular polynomials satisfies τ < 13n13q−3/2.

Conjecture

|τ − q−2| = O(q−3).

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Current work

◮ Exact counting, generating functions (alas, nowhere convergent),

multivariate polynomials (with Alfredo Viola).

◮ Estimates for curves in higher dimensional spaces (with Guillermo

Matera).

◮ Decomposable polynomials.

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Thank you!

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