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  • 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

    ▶ 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

  • 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

    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

    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

  • Results