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 , F q , and some term order on F q [ x 1 , . . . , x r ] . P d = { monic polynomials of degree d in F q [ x 1 , . . . , x r ] } R d = { f ∈ P d : f reducible } M d = { ordered partitions of d } Exact formula r ) − 1 1 − q − ( r − 1+ d r − 1 ) # P d = q ( r + d , 1 − q − 1 ( − 1) | j | µ ( k ) � � # R d = # P d + # P j 1 · # P j 2 · · · # P j | j | . k | j | j ∈ M d/k k | d 19/23
Combinatorics on polynomials: (ir)reducible Fix r ≥ 2 , F q , and some term order on F q [ x 1 , . . . , x r ] . P d = { monic polynomials of degree d in F q [ x 1 , . . . , x r ] } R d = { f ∈ P d : f reducible } M d = { ordered partitions of d } Exact formula r ) − 1 1 − q − ( r − 1+ d r − 1 ) # P d = q ( r + d , 1 − q − 1 ( − 1) | j | µ ( k ) � � # R d = # P d + # P j 1 · # P j 2 · · · # P j | j | . k | j | j ∈ M d/k k | d 19/23
Combinatorics on polynomials: (ir)reducible ) + r − 1 1 − q − r ρ d ( q ) = q ( r + d − 1 r (1 − q − 1 ) 2 Symbolic approximation � � r − 1 ) + r ( r +1) / 2 · 1 + O ( q − r ( r − 1) / 2 ) 1 + q − ( r + d − 2 # R d = ρ d ( q ) · 1 − q − r for d ≥ 4 . Exact formulas for d ≤ 3 . Explicit approximation For d ≥ 4 : | # R d − ρ d ( q ) | ≤ ρ d ( q ) · 3 q − ( r + d − 2 r − 1 ) + r ( r +1) / 2 . 18/23
Combinatorics on polynomials: (ir)reducible ) + r − 1 1 − q − r ρ d ( q ) = q ( r + d − 1 r (1 − q − 1 ) 2 Symbolic approximation � � r − 1 ) + r ( r +1) / 2 · 1 + O ( q − r ( r − 1) / 2 ) 1 + q − ( r + d − 2 # R d = ρ d ( q ) · 1 − q − r for d ≥ 4 . Exact formulas for d ≤ 3 . Explicit approximation For d ≥ 4 : | # R d − ρ d ( q ) | ≤ ρ d ( q ) · 3 q − ( 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 F q . 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 F q . 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 F q . 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
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