Peter Paule 60 years young:cogito ergo summo

Joachim von zur GathenB-IT, Universität Bonn

18 May 2018

Draft (2018Paule60) – May 18, 2018 – 1:23

Peter Paule

▶ ISSAC 1996 Zürich: Peter explains GFF to me for ModernComputer Algebra, very patiently. “Cogito ergo summo”.

▶ Christmas cards, designed by his wife.▶ 2002 Waterloo and Shakespeare at Stratford on Avon.▶ 2011 Christmas salon

21/23

Draft (2018Paule60) – May 18, 2018 – 1:23

Peter Paule

▶ ISSAC 1996 Zürich: Peter explains GFF to me for ModernComputer Algebra, very patiently. “Cogito ergo summo”.

▶ Christmas cards, designed by his wife.▶ 2002 Waterloo and Shakespeare at Stratford on Avon.▶ 2011 Christmas salon

21/23

Draft (2018Paule60) – May 18, 2018 – 1:23

Peter Paule

▶ ISSAC 1996 Zürich: Peter explains GFF to me for ModernComputer Algebra, very patiently. “Cogito ergo summo”.

▶ Christmas cards, designed by his wife.▶ 2002 Waterloo and Shakespeare at Stratford on Avon.▶ 2011 Christmas salon

21/23

Draft (2018Paule60) – May 18, 2018 – 1:23

Peter Paule

▶ ISSAC 1996 Zürich: Peter explains GFF to me for ModernComputer Algebra, very patiently. “Cogito ergo summo”.

▶ Christmas cards, designed by his wife.▶ 2002 Waterloo and Shakespeare at Stratford on Avon.▶ 2011 Christmas salon

21/23

Draft (2018Paule60) – May 18, 2018 – 1:23

Peter Paule

▶ ISSAC 1996 Zürich: Peter explains GFF to me for ModernComputer Algebra, very patiently. “Cogito ergo summo”.

▶ Christmas cards, designed by his wife.▶ 2002 Waterloo and Shakespeare at Stratford on Avon.▶ 2011 Christmas salon

21/23

Draft (2018Paule60) – May 18, 2018 – 1:23

2002 Waterloo.

Peter and my daughter Rafaela on Mark Giesbrecht’s deck.Photo courtesy of Mark Giesbrecht. 20/23

Draft (2018Paule60) – May 18, 2018 – 1:23

From my talk on Alexander von Humboldt at Peter’s 2011 salon:

Cogito ergo summo.

19/23

Draft (2018Paule60) – May 18, 2018 – 1:23

From my talk on Alexander von Humboldt at Peter’s 2011 salon:

Cogito ergo summo.

19/23

Draft (2018Paule60) – May 18, 2018 – 1:23

Technical part: Combinatorics on polynomialequations—do they describe nice varieties?

Joint work with Guillermo Matera

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

18/23

Draft (2018Paule60) – May 18, 2018 – 1:23

Combinatorics on polynomials

General question: given a class of polynomials over finite fields,how many elements does it contain? Equivalent: probability tobe in that class.

Classical: (ir)reducible univariate and multivariate polynomials(Carlitz; Cohen; Wan; Gao & Lauder; Bodin; Hou & Mullen).Amenable to a (non-standard) variant ofgeneratingfunctionology plus some extra work (vzG, Viola &Ziegler). This yields exact formulas, asymptotics, and explicitestimates.

17/23

Draft (2018Paule60) – May 18, 2018 – 1:23

Combinatorics on polynomials

General question: given a class of polynomials over finite fields,how many elements does it contain? Equivalent: probability tobe in that class.

Classical: (ir)reducible univariate and multivariate polynomials(Carlitz; Cohen; Wan; Gao & Lauder; Bodin; Hou & Mullen).Amenable to a (non-standard) variant ofgeneratingfunctionology plus some extra work (vzG, Viola &Ziegler). This yields exact formulas, asymptotics, and explicitestimates.

17/23

Draft (2018Paule60) – May 18, 2018 – 1:23

Combinatorics on polynomials

▶ Irreducibility and other properties for several multivariatepolynomials: this talk. Approximate results.

▶ Previous work: curves in high-dimensional spaces.Approximate results.Model: Chow variety (Eisenbud & Harris; Cesaratto, vzG &Matera).

16/23

Draft (2018Paule60) – May 18, 2018 – 1:23

Combinatorics on polynomials

▶ Irreducibility and other properties for several multivariatepolynomials: this talk. Approximate results.

▶ Previous work: curves in high-dimensional spaces.Approximate results.Model: Chow variety (Eisenbud & Harris; Cesaratto, vzG &Matera).

16/23

Draft (2018Paule60) – May 18, 2018 – 1:23

The task

An algebraic variety V is defined by a system of polynomialequations. A fair number of results in algebraic geometry onlyhold if the system or the variety satisfy certain conditions ofbeing “nice”:

▶ V is a set-theoretic complete intersection. Equivalently:The system is regular, so that no polynomial is a zerodivisor 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 andvarieties.

15/23

Draft (2018Paule60) – May 18, 2018 – 1:23

The task

An algebraic variety V is defined by a system of polynomialequations. A fair number of results in algebraic geometry onlyhold if the system or the variety satisfy certain conditions ofbeing “nice”:

▶ V is a set-theoretic complete intersection. Equivalently:The system is regular, so that no polynomial is a zerodivisor 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 andvarieties.

15/23

Draft (2018Paule60) – May 18, 2018 – 1:23

The task

An algebraic variety V is defined by a system of polynomialequations. A fair number of results in algebraic geometry onlyhold if the system or the variety satisfy certain conditions ofbeing “nice”:

▶ V is a set-theoretic complete intersection. Equivalently:The system is regular, so that no polynomial is a zerodivisor 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 andvarieties.

15/23

Draft (2018Paule60) – May 18, 2018 – 1:23

The task

An algebraic variety V is defined by a system of polynomialequations. A fair number of results in algebraic geometry onlyhold if the system or the variety satisfy certain conditions ofbeing “nice”:

▶ V is a set-theoretic complete intersection. Equivalently:The system is regular, so that no polynomial is a zerodivisor 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 andvarieties.

15/23

Draft (2018Paule60) – May 18, 2018 – 1:23

The task

An algebraic variety V is defined by a system of polynomialequations. A fair number of results in algebraic geometry onlyhold if the system or the variety satisfy certain conditions ofbeing “nice”:

▶ V is a set-theoretic complete intersection. Equivalently:The system is regular, so that no polynomial is a zerodivisor 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 andvarieties.

15/23

Draft (2018Paule60) – May 18, 2018 – 1:23

The task

An algebraic variety V is defined by a system of polynomialequations. A fair number of results in algebraic geometry onlyhold if the system or the variety satisfy certain conditions ofbeing “nice”:

▶ V is a set-theoretic complete intersection. Equivalently:The system is regular, so that no polynomial is a zerodivisor 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 andvarieties.

15/23

Draft (2018Paule60) – May 18, 2018 – 1:23

The task

An algebraic variety V is defined by a system of polynomialequations. A fair number of results in algebraic geometry onlyhold if the system or the variety satisfy certain conditions ofbeing “nice”:

▶ V is a set-theoretic complete intersection. Equivalently:The system is regular, so that no polynomial is a zerodivisor 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 andvarieties.

15/23

Draft (2018Paule60) – May 18, 2018 – 1:23

Results

Setting:▶ field K with algebraic closure K̄,▶ 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

K̄projective variety defined by f over K̄.

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

14/23

Draft (2018Paule60) – May 18, 2018 – 1:23

Results

Setting:▶ field K with algebraic closure K̄,▶ 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

K̄projective variety defined by f over K̄.

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

14/23

Draft (2018Paule60) – May 18, 2018 – 1:23

Results

Setting:▶ field K with algebraic closure K̄,▶ 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

K̄projective variety defined by f over K̄.

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

14/23

Draft (2018Paule60) – May 18, 2018 – 1:23

Results

Setting:▶ field K with algebraic closure K̄,▶ 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

K̄projective variety defined by f over K̄.

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

14/23

Draft (2018Paule60) – May 18, 2018 – 1:23

Results

Setting:▶ field K with algebraic closure K̄,▶ 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

K̄projective variety defined by f over K̄.

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

14/23

Draft (2018Paule60) – May 18, 2018 – 1:23

Results

Setting:▶ field K with algebraic closure K̄,▶ 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

K̄projective variety defined by f over K̄.

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

14/23

Draft (2018Paule60) – May 18, 2018 – 1:23

Results

Setting:▶ field K with algebraic closure K̄,▶ 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

K̄projective variety defined by f over K̄.

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

14/23

Draft (2018Paule60) – May 18, 2018 – 1:23

Results

Setting:▶ field K with algebraic closure K̄,▶ 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

K̄projective variety defined by f over K̄.

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

14/23

Draft (2018Paule60) – May 18, 2018 – 1:23

Results

Setting:▶ field K with algebraic closure K̄,▶ 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

K̄projective variety defined by f over K̄.

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

14/23

Draft (2018Paule60) – May 18, 2018 – 1:23

Results

Setting:▶ field K with algebraic closure K̄,▶ 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

K̄projective variety defined by f over K̄.

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

14/23

Draft (2018Paule60) – May 18, 2018 – 1:23

Results

Setting:▶ field K with algebraic closure K̄,▶ 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

K̄projective variety defined by f over K̄.

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

14/23

Draft (2018Paule60) – May 18, 2018 – 1:23

Results

The first four properties hold for almost all systems, but the lastone, non-degeneracy, does not.

For each of the first four properties “prop”, we find a nonzero“genericity polynomial” Pprop in the coefficients of the systems fwith 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 aprobability that tends rapidly to 1 with growing q.

In contrast, most systems describe a degenerate variety.

13/23

Draft (2018Paule60) – May 18, 2018 – 1:23

Results

The first four properties hold for almost all systems, but the lastone, non-degeneracy, does not.

For each of the first four properties “prop”, we find a nonzero“genericity polynomial” Pprop in the coefficients of the systems fwith 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 aprobability that tends rapidly to 1 with growing q.

In contrast, most systems describe a degenerate variety.

13/23

Draft (2018Paule60) – May 18, 2018 – 1:23

Results

The first four properties hold for almost all systems, but the lastone, non-degeneracy, does not.

For each of the first four properties “prop”, we find a nonzero“genericity polynomial” Pprop in the coefficients of the systems fwith 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 aprobability that tends rapidly to 1 with growing q.

In contrast, most systems describe a degenerate variety.

13/23

Draft (2018Paule60) – May 18, 2018 – 1:23

Results

The first four properties hold for almost all systems, but the lastone, non-degeneracy, does not.

For each of the first four properties “prop”, we find a nonzero“genericity polynomial” Pprop in the coefficients of the systems fwith 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 aprobability that tends rapidly to 1 with growing q.

In contrast, most systems describe a degenerate variety.

13/23

Draft (2018Paule60) – May 18, 2018 – 1:23

Results

The first four properties hold for almost all systems, but the lastone, non-degeneracy, does not.

For each of the first four properties “prop”, we find a nonzero“genericity polynomial” Pprop in the coefficients of the systems fwith 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 aprobability that tends rapidly to 1 with growing q.

In contrast, most systems describe a degenerate variety.

13/23

Draft (2018Paule60) – May 18, 2018 – 1:23

Template of results

For a property “prop”, we state results of the following form.

Geometric theorem. There exists a nonzeromultihomogeneous genericity polynomial Pprop in variablesrepresenting the coefficients of a system f with the followingproperties:

▶ 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 withq ≥ s ·degBound/3, the probability that V (f) has property“prop” for a uniformly random system f over Fq satisfies

1− s ·degBoundq

≤ probability ≤ 1.

Tool: multihomogeneous Weil bounds.12/23

Draft (2018Paule60) – May 18, 2018 – 1:23

Template of results

For a property “prop”, we state results of the following form.

Geometric theorem. There exists a nonzeromultihomogeneous genericity polynomial Pprop in variablesrepresenting the coefficients of a system f with the followingproperties:

▶ 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 withq ≥ s ·degBound/3, the probability that V (f) has property“prop” for a uniformly random system f over Fq satisfies

1− s ·degBoundq

≤ probability ≤ 1.

Tool: multihomogeneous Weil bounds.12/23

Draft (2018Paule60) – May 18, 2018 – 1:23

Template of results

For a property “prop”, we state results of the following form.

Geometric theorem. There exists a nonzeromultihomogeneous genericity polynomial Pprop in variablesrepresenting the coefficients of a system f with the followingproperties:

▶ 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 withq ≥ s ·degBound/3, the probability that V (f) has property“prop” for a uniformly random system f over Fq satisfies

1− s ·degBoundq

≤ probability ≤ 1.

Tool: multihomogeneous Weil bounds.12/23

Draft (2018Paule60) – May 18, 2018 – 1:23

Template of results

For a property “prop”, we state results of the following form.

Geometric theorem. There exists a nonzeromultihomogeneous genericity polynomial Pprop in variablesrepresenting the coefficients of a system f with the followingproperties:

▶ 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 withq ≥ s ·degBound/3, the probability that V (f) has property“prop” for a uniformly random system f over Fq satisfies

1− s ·degBoundq

≤ probability ≤ 1.

Tool: multihomogeneous Weil bounds.12/23

Draft (2018Paule60) – May 18, 2018 – 1:23

Template of results

Feature: degBound and s depend on the geometric systemparameters like r, the degrees, and the property underconsideration, but not on q.

The genericity polynomials are explicitly given and can beevaluated in polynomial time in the model of arithmetic circuits(aka straight-line programs).

11/23

Draft (2018Paule60) – May 18, 2018 – 1:23

Template of results

Feature: degBound and s depend on the geometric systemparameters like r, the degrees, and the property underconsideration, but not on q.

The genericity polynomials are explicitly given and can beevaluated in polynomial time in the model of arithmetic circuits(aka straight-line programs).

11/23

Draft (2018Paule60) – May 18, 2018 – 1:23

Complete intersection

Property: set-theoretic complete intersection. Thus V (f) hasdimension r − s, and (equivalently) the system f forms aregular sequence of K[X0, . . . , Xr].Geometry: degBound = δ.

Property: ideal-theoretic complete intersection. V (f) is aset-theoretic complete intersection and the ideal generated byf in K[X0, . . . , Xr] is radical. In particular, dimV (f) = r− s anddeg V (f) = δ.Geometry:

degBound = 2σδ.

Combinatorics over Fq:

1− 2sσδ

q≤ probability ≤ 1.

10/23

Draft (2018Paule60) – May 18, 2018 – 1:23

Complete intersection

Property: set-theoretic complete intersection. Thus V (f) hasdimension r − s, and (equivalently) the system f forms aregular sequence of K[X0, . . . , Xr].Geometry: degBound = δ.

Property: ideal-theoretic complete intersection. V (f) is aset-theoretic complete intersection and the ideal generated byf in K[X0, . . . , Xr] is radical. In particular, dimV (f) = r− s anddeg V (f) = δ.Geometry:

degBound = 2σδ.

Combinatorics over Fq:

1− 2sσδ

q≤ probability ≤ 1.

10/23

Draft (2018Paule60) – May 18, 2018 – 1:23

Complete intersection

Property: set-theoretic complete intersection. Thus V (f) hasdimension r − s, and (equivalently) the system f forms aregular sequence of K[X0, . . . , Xr].Geometry: degBound = δ.

Property: ideal-theoretic complete intersection. V (f) is aset-theoretic complete intersection and the ideal generated byf in K[X0, . . . , Xr] is radical. In particular, dimV (f) = r− s anddeg V (f) = δ.Geometry:

degBound = 2σδ.

Combinatorics over Fq:

1− 2sσδ

q≤ probability ≤ 1.

10/23

Draft (2018Paule60) – May 18, 2018 – 1:23

Nonsingular complete intersection

Property: V (f) is a nonsingular complete intersection ofdimension 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

Draft (2018Paule60) – May 18, 2018 – 1:23

Nonsingular complete intersection

Property: V (f) is a nonsingular complete intersection ofdimension 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

Draft (2018Paule60) – May 18, 2018 – 1:23

Absolutely irreducible complete intersection

Property: V (f) is an absolutely irreducible completeintersection of dimension r − s and degree δ.Geometry:

degBound = 3σ2δ.

Combinatorics over Fq:

1− 3sσ2δ

q≤ probability ≤ 1.

8/23

Draft (2018Paule60) – May 18, 2018 – 1:23

Absolutely irreducible complete intersection

Property: V (f) is an absolutely irreducible completeintersection of dimension r − s and degree δ.Geometry:

degBound = 3σ2δ.

Combinatorics over Fq:

1− 3sσ2δ

q≤ probability ≤ 1.

8/23

Draft (2018Paule60) – May 18, 2018 – 1:23

Most varieties are degenerate

Fix integers s, b ≥ 2 and consider all d = (d1, . . . , ds) withd1 ≥ d2 ≥ · · · ≥ ds ≥ 1, d1 ≥ 2, and Bézout numberδ(d) = d1 · · · ds = b. These correspond to unorderedfactorizations 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

Draft (2018Paule60) – May 18, 2018 – 1:23

Most varieties are degenerate

Fix integers s, b ≥ 2 and consider all d = (d1, . . . , ds) withd1 ≥ d2 ≥ · · · ≥ ds ≥ 1, d1 ≥ 2, and Bézout numberδ(d) = d1 · · · ds = b. These correspond to unorderedfactorizations 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

Draft (2018Paule60) – May 18, 2018 – 1:23

Most varieties are degenerate

Fix integers s, b ≥ 2 and consider all d = (d1, . . . , ds) withd1 ≥ d2 ≥ · · · ≥ ds ≥ 1, d1 ≥ 2, and Bézout numberδ(d) = d1 · · · ds = b. These correspond to unorderedfactorizations 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

Draft (2018Paule60) – May 18, 2018 – 1:23

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 anabsolutely irreducible complete intersection of dimension r − sand degree b.Then for any d ̸= d(b), we have

dimSd(b) ≥ dimSd + g(b).

6/23

Draft (2018Paule60) – May 18, 2018 – 1:23

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 anabsolutely irreducible complete intersection of dimension r − sand degree b.Then for any d ̸= d(b), we have

dimSd(b) ≥ dimSd + g(b).

6/23

Draft (2018Paule60) – May 18, 2018 – 1:23

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 anabsolutely irreducible hypersurface of dimension r − s anddegree b within some r − s+ 1-dimensional projective linearsubspace, for any d with b = δ(d). Then∣∣∣ N (b)

pDbps−1r

− 1∣∣∣ ≤ 14

qr−s+1+

blog2 log2 b

qg(b).

5/23

Draft (2018Paule60) – May 18, 2018 – 1:23

One proof idea

Property: ideal-theoretic complete intersection. Thepolynomials in f form a regular sequence of K[X0, . . . , Xr] andthe ideal generated by f in K[X0, . . . , Xr] is radical.Geometry:

degBound = 2σδ.

General fact: each irreducible component of V (f1, . . . , fs) hascodimension 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.

4/23

Draft (2018Paule60) – May 18, 2018 – 1:23

One proof idea

Property: ideal-theoretic complete intersection. Thepolynomials in f form a regular sequence of K[X0, . . . , Xr] andthe ideal generated by f in K[X0, . . . , Xr] is radical.Geometry:

degBound = 2σδ.

General fact: each irreducible component of V (f1, . . . , fs) hascodimension 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.

4/23

Draft (2018Paule60) – May 18, 2018 – 1:23

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 isreplaced by a variable. As genericity polynomial we take

P = multivariate resultant of F1, . . . , Fs, J(F ), xs+1, . . . , xr.

The degrees of its arguments in the “coefficient variables” ared1, . . . , ds, σ, 0, . . . , 0. Therefore the degree of P in coeffs(Fi) isd1 · · · 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.

3/23

Draft (2018Paule60) – May 18, 2018 – 1:23

One proof idea

genericity. We take some system f with P (f) ̸= 0. ThenV (f, J(f), xs+1, . . . , xr) is empty. For V ′ = V (f, J(f)), thefollowing hold:

dimV ′ ≤ r − s− 1 =⇒ dimV ′ = r − s− 1

=⇒ dimV (f) = r − s =⇒ (f1, . . . , fs) regular sequence.

Also: the ideal generated by the s× s minors of J(f) hascodimension at least 1 in V (f). Therefore: f generates aradical ideal in K̄[x0, . . . , xr] and V (f) is an ideal-theoreticcomplete intersection. See Eisenbud.

2/23

Draft (2018Paule60) – May 18, 2018 – 1:23

Open questions

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

1/23

Draft (2018Paule60) – May 18, 2018 – 1:23

Open questions

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

1/23

Draft (2018Paule60) – May 18, 2018 – 1:23

Open questions

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

1/23

Draft (2018Paule60) – May 18, 2018 – 1:23

Open questions

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

1/23

Draft (2018Paule60) – May 18, 2018 – 1:23

The end (of this talk)

Alles Gute zum Geburtstag,Peter!

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Draft (2018Paule60) – May 18, 2018 – 1:23