# peter paule 60 years young: cogito ergo summo · pdf file cogito ergo summo joachim von zur...

Post on 16-Aug-2020

0 views

Embed Size (px)

TRANSCRIPT

Peter Paule 60 years young: cogito ergo summo

Joachim von zur Gathen B-IT, Universität Bonn

18 May 2018

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

Peter Paule

▶ ISSAC 1996 Zürich: Peter explains GFF to me for Modern Computer 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 Modern Computer 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 Modern Computer 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

21/23

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

Peter Paule

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 polynomial equations—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 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.

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 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.

17/23

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

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).

16/23

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

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).

16/23

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

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

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

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

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

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

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

The task

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

15/23

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

The task

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

15/23

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

The task

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

15/23

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

The task

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

15/23

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

Results