Combinatorics on polynomial equations: do they describe nice - - PowerPoint PPT Presentation

Combinatorics on polynomial equations: do they describe nice varieties?

Joachim von zur Gathen Bonn Joint work with Guillermo Matera

Overview

◮ Combinatorics on polynomials ◮ Task ◮ Some results ◮ Methods ◮ Open questions

21/23

Combinatorics on polynomials

General question: given a class of polynomials over finite fields, how many elements does it contain? Equivalent: probability to be in that class. Classical: (ir)reducible univariate and multivariate polynomials (Carlitz; Cohen; Wan; Gao & Lauder; Bodin; Hou & Mullen). Amenable to a (non-standard) variant of generatingfunctionology plus some extra work (vzG, Viola & Ziegler). This yields exact formulas, asymptotics, and explicit estimates.

20/23

Combinatorics on polynomials

General question: given a class of polynomials over finite fields, how many elements does it contain? Equivalent: probability to be in that class. Classical: (ir)reducible univariate and multivariate polynomials (Carlitz; Cohen; Wan; Gao & Lauder; Bodin; Hou & Mullen). Amenable to a (non-standard) variant of generatingfunctionology plus some extra work (vzG, Viola & Ziegler). This yields exact formulas, asymptotics, and explicit estimates.

20/23

Combinatorics on polynomials: (ir)reducible

Fix r ≥ 2, Fq, and some term order on Fq[x1, . . . , xr]. Pd = {monic polynomials of degree d in Fq[x1, . . . , xr]} Rd = {f ∈ Pd : f reducible} Md = {ordered partitions of d}

Exact formula

#Pd = q(r+d

r )−1 1 − q−(r−1+d r−1 )

1 − q−1 , #Rd = #Pd +

• k|d

µ(k) k

• j∈Md/k

(−1)|j| |j| #Pj1 · #Pj2 · · · #Pj|j|.

19/23

Combinatorics on polynomials: (ir)reducible

Fix r ≥ 2, Fq, and some term order on Fq[x1, . . . , xr]. Pd = {monic polynomials of degree d in Fq[x1, . . . , xr]} Rd = {f ∈ Pd : f reducible} Md = {ordered partitions of d}

Exact formula

#Pd = q(r+d

r )−1 1 − q−(r−1+d r−1 )

1 − q−1 , #Rd = #Pd +

• k|d

µ(k) k

• j∈Md/k

(−1)|j| |j| #Pj1 · #Pj2 · · · #Pj|j|.

19/23

Combinatorics on polynomials: (ir)reducible

ρd(q) = q(r+d−1

r

)+r−1 1 − q−r (1 − q−1)2

Symbolic approximation

#Rd = ρd(q) ·

• 1 + q−(r+d−2

r−1 )+r(r+1)/2 · 1 + O(q−r(r−1)/2)

1 − q−r

• for d ≥ 4. Exact formulas for d ≤ 3.

Explicit approximation

For d ≥ 4: |#Rd − ρd(q)| ≤ ρd(q) · 3q−(r+d−2

r−1 )+r(r+1)/2. 18/23

Combinatorics on polynomials: (ir)reducible

ρd(q) = q(r+d−1

r

)+r−1 1 − q−r (1 − q−1)2

Symbolic approximation

#Rd = ρd(q) ·

• 1 + q−(r+d−2

r−1 )+r(r+1)/2 · 1 + O(q−r(r−1)/2)

1 − q−r

• for d ≥ 4. Exact formulas for d ≤ 3.

Explicit approximation

For d ≥ 4: |#Rd − ρd(q)| ≤ ρd(q) · 3q−(r+d−2

r−1 )+r(r+1)/2. 18/23

Combinatorics on polynomials

Similar: s-powerful, relatively and absolutely irreducible, coprime pairs of polynomials. Smooth bivariate polynomials. Decomposable univariate polynomials. Long and distinguished history, starting with Ritt, Fatou, and Julia in the 1920s. Dichotomy tame vs. wild (characteristic p does not divide the degree vs. does). Exact counting in the tame case: Ziegler. Wild case: partial results, not completely understood. Additive polynomials: Giesbrecht; vzG, Giesbrecht & Ziegler. Special case of Ore polynomials. Degree p composed with degree p: Blankertz, vzG & Ziegler.

17/23

Combinatorics on polynomials

Similar: s-powerful, relatively and absolutely irreducible, coprime pairs of polynomials. Smooth bivariate polynomials. Decomposable univariate polynomials. Long and distinguished history, starting with Ritt, Fatou, and Julia in the 1920s. Dichotomy tame vs. wild (characteristic p does not divide the degree vs. does). Exact counting in the tame case: Ziegler. Wild case: partial results, not completely understood. Additive polynomials: Giesbrecht; vzG, Giesbrecht & Ziegler. Special case of Ore polynomials. Degree p composed with degree p: Blankertz, vzG & Ziegler.

17/23

Combinatorics on polynomials

Similar: s-powerful, relatively and absolutely irreducible, coprime pairs of polynomials. Smooth bivariate polynomials. Decomposable univariate polynomials. Long and distinguished history, starting with Ritt, Fatou, and Julia in the 1920s. Dichotomy tame vs. wild (characteristic p does not divide the degree vs. does). Exact counting in the tame case: Ziegler. Wild case: partial results, not completely understood. Additive polynomials: Giesbrecht; vzG, Giesbrecht & Ziegler. Special case of Ore polynomials. Degree p composed with degree p: Blankertz, vzG & Ziegler.

17/23

Combinatorics on polynomials

◮ Irreducibility and other properties for several multivariate

polynomials: this talk. Approximate results.

◮ Previous work: curves in high-dimensional spaces.

Approximate results. Model: Chow variety (Eisenbud & Harris; Cesaratto, vzG & Matera). In all instances, one obtains a polynomial in q that describes the number of elements over Fq. Exception: degree p composed with degree p.

16/23

Combinatorics on polynomials

◮ Irreducibility and other properties for several multivariate

polynomials: this talk. Approximate results.

◮ Previous work: curves in high-dimensional spaces.

Approximate results. Model: Chow variety (Eisenbud & Harris; Cesaratto, vzG & Matera). In all instances, one obtains a polynomial in q that describes the number of elements over Fq. Exception: degree p composed with degree p.

16/23

Combinatorics on polynomials

◮ Irreducibility and other properties for several multivariate

polynomials: this talk. Approximate results.

◮ Previous work: curves in high-dimensional spaces.

Approximate results. Model: Chow variety (Eisenbud & Harris; Cesaratto, vzG & Matera). In all instances, one obtains a polynomial in q that describes the number of elements over Fq. Exception: degree p composed with degree p.

16/23

The task

An algebraic variety V is defined by a system of polynomial

• equations. A fair number of results in algebraic geometry only

hold if the system or the variety satisfy certain conditions of being “nice”:

◮ V is a set-theoretic complete intersection. Equivalently:

The system is regular, so that no polynomial is a zero divisor modulo the previous ones.

◮ V is an ideal-theoretic complete intersection. ◮ V is absolutely irreducible. ◮ V is nonsingular. ◮ V is non-degenerate (not contained in a hyperplane).

Intuition: these five properties hold for most systems and varieties.

15/23

The task

An algebraic variety V is defined by a system of polynomial

• equations. A fair number of results in algebraic geometry only

hold if the system or the variety satisfy certain conditions of being “nice”:

◮ V is a set-theoretic complete intersection. Equivalently:

The system is regular, so that no polynomial is a zero divisor modulo the previous ones.

◮ V is an ideal-theoretic complete intersection. ◮ V is absolutely irreducible. ◮ V is nonsingular. ◮ V is non-degenerate (not contained in a hyperplane).

Intuition: these five properties hold for most systems and varieties.

15/23

The task

An algebraic variety V is defined by a system of polynomial

• equations. A fair number of results in algebraic geometry only

hold if the system or the variety satisfy certain conditions of being “nice”:

◮ V is a set-theoretic complete intersection. Equivalently:

The system is regular, so that no polynomial is a zero divisor modulo the previous ones.

◮ V is an ideal-theoretic complete intersection. ◮ V is absolutely irreducible. ◮ V is nonsingular. ◮ V is non-degenerate (not contained in a hyperplane).

Intuition: these five properties hold for most systems and varieties.

15/23

The task

An algebraic variety V is defined by a system of polynomial

• equations. A fair number of results in algebraic geometry only

hold if the system or the variety satisfy certain conditions of being “nice”:

◮ V is a set-theoretic complete intersection. Equivalently:

The system is regular, so that no polynomial is a zero divisor modulo the previous ones.

◮ V is an ideal-theoretic complete intersection. ◮ V is absolutely irreducible. ◮ V is nonsingular. ◮ V is non-degenerate (not contained in a hyperplane).

Intuition: these five properties hold for most systems and varieties.

15/23

The task

An algebraic variety V is defined by a system of polynomial

• equations. A fair number of results in algebraic geometry only

hold if the system or the variety satisfy certain conditions of being “nice”:

◮ V is a set-theoretic complete intersection. Equivalently:

The system is regular, so that no polynomial is a zero divisor modulo the previous ones.

◮ V is an ideal-theoretic complete intersection. ◮ V is absolutely irreducible. ◮ V is nonsingular. ◮ V is non-degenerate (not contained in a hyperplane).

Intuition: these five properties hold for most systems and varieties.

15/23

The task

An algebraic variety V is defined by a system of polynomial

• equations. A fair number of results in algebraic geometry only

hold if the system or the variety satisfy certain conditions of being “nice”:

◮ V is a set-theoretic complete intersection. Equivalently:

The system is regular, so that no polynomial is a zero divisor modulo the previous ones.

◮ V is an ideal-theoretic complete intersection. ◮ V is absolutely irreducible. ◮ V is nonsingular. ◮ V is non-degenerate (not contained in a hyperplane).

Intuition: these five properties hold for most systems and varieties.

15/23

The task

An algebraic variety V is defined by a system of polynomial

• equations. A fair number of results in algebraic geometry only

hold if the system or the variety satisfy certain conditions of being “nice”:

◮ V is a set-theoretic complete intersection. Equivalently:

The system is regular, so that no polynomial is a zero divisor modulo the previous ones.

◮ V is an ideal-theoretic complete intersection. ◮ V is absolutely irreducible. ◮ V is nonsingular. ◮ V is non-degenerate (not contained in a hyperplane).

Intuition: these five properties hold for most systems and varieties.

15/23

Results

Setting:

◮ field K with algebraic closure ¯

K,

◮ projective varieties and homogeneous polynomials, ◮ r = dimension = (number of variables) −1, ◮ s = number of polynomials, ◮ d = (d1, . . . , ds) degree pattern, ◮ f = (f1, . . . , fs) system of homogeneous polynomials in

r + 1 variables over K with deg fi = di for all i,

◮ δ = d1 · · · ds = Bézout number, ◮ σ = d1 + · · · + ds − s = “sum” of degrees, ◮ V (f) ⊆ Pr ¯ K projective variety defined by f over ¯

K. The set of all such f forms a multiprojective space over K.

14/23

Results

Setting:

◮ field K with algebraic closure ¯

K,

◮ projective varieties and homogeneous polynomials, ◮ r = dimension = (number of variables) −1, ◮ s = number of polynomials, ◮ d = (d1, . . . , ds) degree pattern, ◮ f = (f1, . . . , fs) system of homogeneous polynomials in

r + 1 variables over K with deg fi = di for all i,

◮ δ = d1 · · · ds = Bézout number, ◮ σ = d1 + · · · + ds − s = “sum” of degrees, ◮ V (f) ⊆ Pr ¯ K projective variety defined by f over ¯

K. The set of all such f forms a multiprojective space over K.

14/23

Results

Setting:

◮ field K with algebraic closure ¯

K,

◮ projective varieties and homogeneous polynomials, ◮ r = dimension = (number of variables) −1, ◮ s = number of polynomials, ◮ d = (d1, . . . , ds) degree pattern, ◮ f = (f1, . . . , fs) system of homogeneous polynomials in

r + 1 variables over K with deg fi = di for all i,

◮ δ = d1 · · · ds = Bézout number, ◮ σ = d1 + · · · + ds − s = “sum” of degrees, ◮ V (f) ⊆ Pr ¯ K projective variety defined by f over ¯

K. The set of all such f forms a multiprojective space over K.

14/23

Results

Setting:

◮ field K with algebraic closure ¯

K,

◮ projective varieties and homogeneous polynomials, ◮ r = dimension = (number of variables) −1, ◮ s = number of polynomials, ◮ d = (d1, . . . , ds) degree pattern, ◮ f = (f1, . . . , fs) system of homogeneous polynomials in

r + 1 variables over K with deg fi = di for all i,

◮ δ = d1 · · · ds = Bézout number, ◮ σ = d1 + · · · + ds − s = “sum” of degrees, ◮ V (f) ⊆ Pr ¯ K projective variety defined by f over ¯

K. The set of all such f forms a multiprojective space over K.

14/23

Results

Setting:

◮ field K with algebraic closure ¯

K,

◮ projective varieties and homogeneous polynomials, ◮ r = dimension = (number of variables) −1, ◮ s = number of polynomials, ◮ d = (d1, . . . , ds) degree pattern, ◮ f = (f1, . . . , fs) system of homogeneous polynomials in

r + 1 variables over K with deg fi = di for all i,

◮ δ = d1 · · · ds = Bézout number, ◮ σ = d1 + · · · + ds − s = “sum” of degrees, ◮ V (f) ⊆ Pr ¯ K projective variety defined by f over ¯

K. The set of all such f forms a multiprojective space over K.

14/23

Results

Setting:

◮ field K with algebraic closure ¯

K,

◮ projective varieties and homogeneous polynomials, ◮ r = dimension = (number of variables) −1, ◮ s = number of polynomials, ◮ d = (d1, . . . , ds) degree pattern, ◮ f = (f1, . . . , fs) system of homogeneous polynomials in

r + 1 variables over K with deg fi = di for all i,

◮ δ = d1 · · · ds = Bézout number, ◮ σ = d1 + · · · + ds − s = “sum” of degrees, ◮ V (f) ⊆ Pr ¯ K projective variety defined by f over ¯

K. The set of all such f forms a multiprojective space over K.

14/23

Results

Setting:

◮ field K with algebraic closure ¯

K,

◮ projective varieties and homogeneous polynomials, ◮ r = dimension = (number of variables) −1, ◮ s = number of polynomials, ◮ d = (d1, . . . , ds) degree pattern, ◮ f = (f1, . . . , fs) system of homogeneous polynomials in

r + 1 variables over K with deg fi = di for all i,

◮ δ = d1 · · · ds = Bézout number, ◮ σ = d1 + · · · + ds − s = “sum” of degrees, ◮ V (f) ⊆ Pr ¯ K projective variety defined by f over ¯

K. The set of all such f forms a multiprojective space over K.

14/23

Results

Setting:

◮ field K with algebraic closure ¯

K,

◮ projective varieties and homogeneous polynomials, ◮ r = dimension = (number of variables) −1, ◮ s = number of polynomials, ◮ d = (d1, . . . , ds) degree pattern, ◮ f = (f1, . . . , fs) system of homogeneous polynomials in

r + 1 variables over K with deg fi = di for all i,

◮ δ = d1 · · · ds = Bézout number, ◮ σ = d1 + · · · + ds − s = “sum” of degrees, ◮ V (f) ⊆ Pr ¯ K projective variety defined by f over ¯

K. The set of all such f forms a multiprojective space over K.

14/23

Results

Setting:

◮ field K with algebraic closure ¯

K,

◮ projective varieties and homogeneous polynomials, ◮ r = dimension = (number of variables) −1, ◮ s = number of polynomials, ◮ d = (d1, . . . , ds) degree pattern, ◮ f = (f1, . . . , fs) system of homogeneous polynomials in

r + 1 variables over K with deg fi = di for all i,

◮ δ = d1 · · · ds = Bézout number, ◮ σ = d1 + · · · + ds − s = “sum” of degrees, ◮ V (f) ⊆ Pr ¯ K projective variety defined by f over ¯

K. The set of all such f forms a multiprojective space over K.

14/23

Results

Setting:

◮ field K with algebraic closure ¯

K,

◮ projective varieties and homogeneous polynomials, ◮ r = dimension = (number of variables) −1, ◮ s = number of polynomials, ◮ d = (d1, . . . , ds) degree pattern, ◮ f = (f1, . . . , fs) system of homogeneous polynomials in

r + 1 variables over K with deg fi = di for all i,

◮ δ = d1 · · · ds = Bézout number, ◮ σ = d1 + · · · + ds − s = “sum” of degrees, ◮ V (f) ⊆ Pr ¯ K projective variety defined by f over ¯

K. The set of all such f forms a multiprojective space over K.

14/23

Results

Setting:

◮ field K with algebraic closure ¯

K,

◮ projective varieties and homogeneous polynomials, ◮ r = dimension = (number of variables) −1, ◮ s = number of polynomials, ◮ d = (d1, . . . , ds) degree pattern, ◮ f = (f1, . . . , fs) system of homogeneous polynomials in

r + 1 variables over K with deg fi = di for all i,

◮ δ = d1 · · · ds = Bézout number, ◮ σ = d1 + · · · + ds − s = “sum” of degrees, ◮ V (f) ⊆ Pr ¯ K projective variety defined by f over ¯

K. The set of all such f forms a multiprojective space over K.

14/23

Results

The first four properties hold for almost all systems, but the last

• ne, non-degeneracy, does not.

For each of the first four properties “prop”, we find a nonzero “obstruction polynomial” Pprop in the coefficients of the systems f with explicitly bounded degree so that for all systems f we have Pprop(f) = 0 = ⇒ V (f) has the property. When K = Fq is a finite field, then each property holds with a probability that tends rapidly to 1 with growing q. In contrast, most systems describe a degenerate variety.

13/23

Results

The first four properties hold for almost all systems, but the last

• ne, non-degeneracy, does not.

For each of the first four properties “prop”, we find a nonzero “obstruction polynomial” Pprop in the coefficients of the systems f with explicitly bounded degree so that for all systems f we have Pprop(f) = 0 = ⇒ V (f) has the property. When K = Fq is a finite field, then each property holds with a probability that tends rapidly to 1 with growing q. In contrast, most systems describe a degenerate variety.

13/23

Results

The first four properties hold for almost all systems, but the last

• ne, non-degeneracy, does not.

For each of the first four properties “prop”, we find a nonzero “obstruction polynomial” Pprop in the coefficients of the systems f with explicitly bounded degree so that for all systems f we have Pprop(f) = 0 = ⇒ V (f) has the property. When K = Fq is a finite field, then each property holds with a probability that tends rapidly to 1 with growing q. In contrast, most systems describe a degenerate variety.

13/23

Results

The first four properties hold for almost all systems, but the last

• ne, non-degeneracy, does not.

For each of the first four properties “prop”, we find a nonzero “obstruction polynomial” Pprop in the coefficients of the systems f with explicitly bounded degree so that for all systems f we have Pprop(f) = 0 = ⇒ V (f) has the property. When K = Fq is a finite field, then each property holds with a probability that tends rapidly to 1 with growing q. In contrast, most systems describe a degenerate variety.

13/23

Results

The first four properties hold for almost all systems, but the last

• ne, non-degeneracy, does not.

For each of the first four properties “prop”, we find a nonzero “obstruction polynomial” Pprop in the coefficients of the systems f with explicitly bounded degree so that for all systems f we have Pprop(f) = 0 = ⇒ V (f) has the property. When K = Fq is a finite field, then each property holds with a probability that tends rapidly to 1 with growing q. In contrast, most systems describe a degenerate variety.

13/23

Template of results

For a property “prop”, we state results of the following form. Geometric theorem. There exists a nonzero multihomogeneous obstruction polynomial Pprop in variables representing the coefficients of a system f with the following properties:

◮ For each f with Pprop(f) = 0, V (f) has property “prop”. ◮ The degree of Pprop in each of the s sets of variables is at

most degBound. Combinatorial corollary. For a finite field Fq with q ≥ s · degBound/3, the probability that V (f) has property “prop” for a uniformly random system f over Fq satisfies 1 − s · degBound q ≤ probability ≤ 1. Tool: multihomogeneous Weil bounds.

12/23

Template of results

For a property “prop”, we state results of the following form. Geometric theorem. There exists a nonzero multihomogeneous obstruction polynomial Pprop in variables representing the coefficients of a system f with the following properties:

◮ For each f with Pprop(f) = 0, V (f) has property “prop”. ◮ The degree of Pprop in each of the s sets of variables is at

most degBound. Combinatorial corollary. For a finite field Fq with q ≥ s · degBound/3, the probability that V (f) has property “prop” for a uniformly random system f over Fq satisfies 1 − s · degBound q ≤ probability ≤ 1. Tool: multihomogeneous Weil bounds.

12/23

Template of results

For a property “prop”, we state results of the following form. Geometric theorem. There exists a nonzero multihomogeneous obstruction polynomial Pprop in variables representing the coefficients of a system f with the following properties:

◮ For each f with Pprop(f) = 0, V (f) has property “prop”. ◮ The degree of Pprop in each of the s sets of variables is at

most degBound. Combinatorial corollary. For a finite field Fq with q ≥ s · degBound/3, the probability that V (f) has property “prop” for a uniformly random system f over Fq satisfies 1 − s · degBound q ≤ probability ≤ 1. Tool: multihomogeneous Weil bounds.

12/23

Template of results

For a property “prop”, we state results of the following form. Geometric theorem. There exists a nonzero multihomogeneous obstruction polynomial Pprop in variables representing the coefficients of a system f with the following properties:

◮ For each f with Pprop(f) = 0, V (f) has property “prop”. ◮ The degree of Pprop in each of the s sets of variables is at

most degBound. Combinatorial corollary. For a finite field Fq with q ≥ s · degBound/3, the probability that V (f) has property “prop” for a uniformly random system f over Fq satisfies 1 − s · degBound q ≤ probability ≤ 1. Tool: multihomogeneous Weil bounds.

12/23

Template of results

Feature: degBound and s depend on the geometric system parameters like r, the degrees, and the property under consideration, but not on q. The obstruction polynomials are explicitly given and can be evaluated in polynomial time in the model of arithmetic circuits (aka straight-line programs).

11/23

Template of results

Feature: degBound and s depend on the geometric system parameters like r, the degrees, and the property under consideration, but not on q. The obstruction polynomials are explicitly given and can be evaluated in polynomial time in the model of arithmetic circuits (aka straight-line programs).

11/23

Complete intersection

Property: set-theoretic complete intersection. Thus V (f) has dimension r − s, and (equivalently) the system f forms a regular sequence of K[X0, . . . , Xr]. Geometry: degBound = δ. Property: ideal-theoretic complete intersection. V (f) is a set-theoretic complete intersection and the ideal generated by f in K[X0, . . . , Xr] is radical. In particular, dim V (f) = r − s and deg V (f) = δ. Geometry: degBound = 2σδ. Combinatorics over Fq: 1 − 2sσδ q ≤ probability ≤ 1.

10/23

Complete intersection

Property: set-theoretic complete intersection. Thus V (f) has dimension r − s, and (equivalently) the system f forms a regular sequence of K[X0, . . . , Xr]. Geometry: degBound = δ. Property: ideal-theoretic complete intersection. V (f) is a set-theoretic complete intersection and the ideal generated by f in K[X0, . . . , Xr] is radical. In particular, dim V (f) = r − s and deg V (f) = δ. Geometry: degBound = 2σδ. Combinatorics over Fq: 1 − 2sσδ q ≤ probability ≤ 1.

10/23

Complete intersection

Property: set-theoretic complete intersection. Thus V (f) has dimension r − s, and (equivalently) the system f forms a regular sequence of K[X0, . . . , Xr]. Geometry: degBound = δ. Property: ideal-theoretic complete intersection. V (f) is a set-theoretic complete intersection and the ideal generated by f in K[X0, . . . , Xr] is radical. In particular, dim V (f) = r − s and deg V (f) = δ. Geometry: degBound = 2σδ. Combinatorics over Fq: 1 − 2sσδ q ≤ probability ≤ 1.

10/23

Nonsingular complete intersection

Property: V (f) is a nonsingular complete intersection of dimension r − s and degree δ. Geometry: degBound = (σ + r)σr−sδ. Combinatorics over Fq: 1 − s(σ + r)σr−sδ q ≤ probability ≤ 1.

9/23

Nonsingular complete intersection

Property: V (f) is a nonsingular complete intersection of dimension r − s and degree δ. Geometry: degBound = (σ + r)σr−sδ. Combinatorics over Fq: 1 − s(σ + r)σr−sδ q ≤ probability ≤ 1.

9/23

Absolutely irreducible complete intersection

Property: V (f) is an absolutely irreducible complete intersection of dimension r − s and degree δ. Geometry: degBound = 3σ2δ. Combinatorics over Fq: 1 − 3sσ2δ q ≤ probability ≤ 1.

8/23

Absolutely irreducible complete intersection

Property: V (f) is an absolutely irreducible complete intersection of dimension r − s and degree δ. Geometry: degBound = 3σ2δ. Combinatorics over Fq: 1 − 3sσ2δ q ≤ probability ≤ 1.

8/23

Most varieties are degenerate

Fix integers s, b ≥ 2 and consider all d = (d1, . . . , ds) with d1 ≥ d2 ≥ · · · ≥ ds ≥ 1, d1 ≥ 2, and Bézout number δ(d) = d1 · · · ds = b. These correspond to unordered factorizations of b, 1 being allowed as a factor.

• Lemma. The number of all such d is at most blog2 log2 b.

Notation: Di(d) = r + di r

• − 1

for 1 ≤ i ≤ s, D(d) = (D1(d), . . . , Ds(d)), |D(d)| = D1(d) + · · · + Ds(d), d(b) = (b, 1, . . . , 1), D(b) = D(b(d)).

7/23

Most varieties are degenerate

Fix integers s, b ≥ 2 and consider all d = (d1, . . . , ds) with d1 ≥ d2 ≥ · · · ≥ ds ≥ 1, d1 ≥ 2, and Bézout number δ(d) = d1 · · · ds = b. These correspond to unordered factorizations of b, 1 being allowed as a factor.

• Lemma. The number of all such d is at most blog2 log2 b.

Notation: Di(d) = r + di r

• − 1

for 1 ≤ i ≤ s, D(d) = (D1(d), . . . , Ds(d)), |D(d)| = D1(d) + · · · + Ds(d), d(b) = (b, 1, . . . , 1), D(b) = D(b(d)).

7/23

Most varieties are degenerate

Fix integers s, b ≥ 2 and consider all d = (d1, . . . , ds) with d1 ≥ d2 ≥ · · · ≥ ds ≥ 1, d1 ≥ 2, and Bézout number δ(d) = d1 · · · ds = b. These correspond to unordered factorizations of b, 1 being allowed as a factor.

• Lemma. The number of all such d is at most blog2 log2 b.

Notation: Di(d) = r + di r

• − 1

for 1 ≤ i ≤ s, D(d) = (D1(d), . . . , Ds(d)), |D(d)| = D1(d) + · · · + Ds(d), d(b) = (b, 1, . . . , 1), D(b) = D(b(d)).

7/23

Most varieties are degenerate

• Lemma. For all d = d(b), we have |D(b)| ≥ |D(b)| + g(b), where

g(b) = b + r r

• − 2

b/2 + r r

• .

Then g(b) ≥ 1.

• Geometry. S(d) = set of all f with degree pattern d, defining an

absolutely irreducible complete intersection of dimension r − s and degree b. Then for any d = d(b), we have dim Sd(b) ≥ dim Sd + g(b).

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Most varieties are degenerate

• Lemma. For all d = d(b), we have |D(b)| ≥ |D(b)| + g(b), where

g(b) = b + r r

• − 2

b/2 + r r

• .

Then g(b) ≥ 1.

• Geometry. S(d) = set of all f with degree pattern d, defining an

absolutely irreducible complete intersection of dimension r − s and degree b. Then for any d = d(b), we have dim Sd(b) ≥ dim Sd + g(b).

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Most varieties are degenerate

Combinatorics over Fq. pr = #Pr(Fq) = qr + qr−1 + · · · + 1. The number of all f with degree pattern d(b) is pDb · ps−1

r

. N (b) = number of polynomial sequences over Fq defining an absolutely irreducible hypersurface of dimension r − s and degree b within some r − s + 1-dimensional projective linear subspace, for any d with b = δ(d). Then

• N (b)

pDbps−1

r

− 1

• ≤

14 qr−s+1 + blog2 log2 b qg(b) .

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One proof idea

Property: ideal-theoretic complete intersection. The polynomials in f form a regular sequence of K[X0, . . . , Xr] and the ideal generated by f in K[X0, . . . , Xr] is radical. Geometry: degBound = 2σδ. General fact: each irreducible component of V (f1, . . . , fs) has codimension at most s. If s = r + 1, then “typically” V (f0, . . . , fr) is empty. If this is not the case, then the resultant of (f0, . . . , fr, J(f)) vanishes, where J(f0, . . . , fr) = det

• ( ∂fi

∂xj )0≤i,j≤r

• is the determinant of the Jacobian.

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One proof idea

Property: ideal-theoretic complete intersection. The polynomials in f form a regular sequence of K[X0, . . . , Xr] and the ideal generated by f in K[X0, . . . , Xr] is radical. Geometry: degBound = 2σδ. General fact: each irreducible component of V (f1, . . . , fs) has codimension at most s. If s = r + 1, then “typically” V (f0, . . . , fr) is empty. If this is not the case, then the resultant of (f0, . . . , fr, J(f)) vanishes, where J(f0, . . . , fr) = det

• ( ∂fi

∂xj )0≤i,j≤r

• is the determinant of the Jacobian.

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One proof idea

Back to f = (f1, . . . , fs). Each fi is a sum of terms coefficient · power product of x0, . . . , xr. We consider F = (F1, . . . , Fs), where each such coefficient is replaced by a variable. As obstruction polynomial we take P = multivariate resultant of F1, . . . , Fs, J(F), xs+1, . . . , xr. The degrees of its arguments in the “coefficient variables” are d1, . . . , ds, σ, 0, . . . , 0. Therefore the degree of P in coeffs(Fi) is d1 · · · di−1di+1 · · · dsσ = σδ/di from F1, . . . , Fs, plus δ from J(F). The total comes to σδ/di + δ ≤ 2σδ. See Cox, Little, O’Shea.

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One proof idea

• Obstruction. We take some system f with P(f) = 0. Then

V (f, J(f), xs+1, . . . , xr) is empty. For V ′ = V (f, J(f)), the following hold: dim V ′ ≤ r − s − 1 = ⇒ dim V ′ = r − s − 1 = ⇒ dim V (f) = r − s = ⇒ (f1, . . . , fs) regular sequence. Also: the ideal generated by the s × s minors of J(f) has codimension at least 1 in V (f). Therefore: f generates a radical ideal in ¯ K[x0, . . . , xr] and V (f) is an ideal-theoretic complete intersection. See Eisenbud.

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Open questions

◮ Affine varieties. ◮ More precise obstructions: necessary and sufficient. ◮ Relation between this model and Chow model for varieties. ◮ Another “nice”: Gröbner basis in singly-exponential time.

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Open questions

◮ Affine varieties. ◮ More precise obstructions: necessary and sufficient. ◮ Relation between this model and Chow model for varieties. ◮ Another “nice”: Gröbner basis in singly-exponential time.

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Open questions

◮ Affine varieties. ◮ More precise obstructions: necessary and sufficient. ◮ Relation between this model and Chow model for varieties. ◮ Another “nice”: Gröbner basis in singly-exponential time.

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Open questions

◮ Affine varieties. ◮ More precise obstructions: necessary and sufficient. ◮ Relation between this model and Chow model for varieties. ◮ Another “nice”: Gröbner basis in singly-exponential time.

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The end

Thank you

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