Proofs by example Benjamin Matschke Boston University Number - PowerPoint PPT Presentation

Proofs by example Benjamin Matschke Boston University Number Theory Seminar Harvard, Oct. 2019

P ROOFS BY EXAMPLE Proofs by example

P ROOFS BY EXAMPLE � To prove a general statement by verifying it for a single example.

P ROOFS BY EXAMPLE � To prove a general statement by verifying it for a single example. For instance: Statement: “All primes are even.”

P ROOFS BY EXAMPLE � To prove a general statement by verifying it for a single example. For instance: Statement: “All primes are even.” Example: 2.

P ROOFS BY EXAMPLE � To prove a general statement by verifying it for a single example. For instance: Statement: “All primes are even.” Example: 2. Wikipedia: “Proof by example” = inappropriate generalization

P ROOFS BY EXAMPLE � To prove a general statement by verifying it for a single example. For instance: Statement: “All primes are even.” Example: 2. Wikipedia: “Proof by example” = inappropriate generalization = logical fallacy, in which one or more examples are claimed as “proof” for a more general statement.

P ROOFS BY EXAMPLE � To prove a general statement by verifying it for a single example. For instance: Statement: “All primes are even.” Example: 2. Wikipedia: “Proof by example” = inappropriate generalization = logical fallacy, in which one or more examples are claimed as “proof” for a more general statement. Related to “law of small numbers”:

P ROOFS BY EXAMPLE � To prove a general statement by verifying it for a single example. For instance: Statement: “All primes are even.” Example: 2. Wikipedia: “Proof by example” = inappropriate generalization = logical fallacy, in which one or more examples are claimed as “proof” for a more general statement. Related to “law of small numbers”: Initial data points do not always predict the subsequent ones.

P ROOFS BY EXAMPLE � To prove a general statement by verifying it for a single example. For instance: Statement: “All primes are even.” Example: 2. Wikipedia: “Proof by example” = inappropriate generalization = logical fallacy, in which one or more examples are claimed as “proof” for a more general statement. Related to “law of small numbers”: Initial data points do not always predict the subsequent ones. Example: 1 , 1 , 2 , 3 , 5 , 8 , 13 , . . . ?

P ROOFS BY EXAMPLE Another example: Thales’ theorem Thales of Miletus ∼ 600 BC

P ROOFS BY EXAMPLE Another example: Thales’ theorem Thales of Miletus ∼ 600 BC

P ROOFS BY EXAMPLE Another example: Thales’ theorem Thales of Miletus ∼ 600 BC

P ROOFS BY EXAMPLE Another example: Thales’ theorem Thales of Miletus � Can “Proof by example” work? ∼ 600 BC

❈ P ROOFS BY EXAMPLE Algebraic setting

❈ P ROOFS BY EXAMPLE Algebraic setting (first attempt):

P ROOFS BY EXAMPLE Algebraic setting (first attempt): Let X = V ( f 1 , . . . , f m ) ⊆ ❈ n be algebraic variety, dim X = d .

P ROOFS BY EXAMPLE Algebraic setting (first attempt): Let X = V ( f 1 , . . . , f m ) ⊆ ❈ n be algebraic variety, dim X = d . Let g ( x 1 , . . . , x n ) be polynomial.

P ROOFS BY EXAMPLE Algebraic setting (first attempt): Let X = V ( f 1 , . . . , f m ) ⊆ ❈ n be algebraic variety, dim X = d . Let g ( x 1 , . . . , x n ) be polynomial. Call P ∈ X “sufficiently generic” for g if g ( P ) = 0 = ⇒ g | X = 0 .

P ROOFS BY EXAMPLE Algebraic setting (first attempt): Let X = V ( f 1 , . . . , f m ) ⊆ ❈ n be algebraic variety, dim X = d . Let g ( x 1 , . . . , x n ) be polynomial. Call P ∈ X “sufficiently generic” for g if g ( P ) = 0 = ⇒ g | X = 0 .

P ROOFS BY EXAMPLE Algebraic setting (first attempt): Let X = V ( f 1 , . . . , f m ) ⊆ ❈ n be algebraic variety, dim X = d . Let g ( x 1 , . . . , x n ) be polynomial. Call P ∈ X “sufficiently generic” for g if g ( P ) = 0 = ⇒ g | X = 0 . Example: Let ∗ be the generic point of X in scheme theoretic sense .

P ROOFS BY EXAMPLE Algebraic setting (first attempt): Let X = V ( f 1 , . . . , f m ) ⊆ ❈ n be algebraic variety, dim X = d . Let g ( x 1 , . . . , x n ) be polynomial. Call P ∈ X “sufficiently generic” for g if g ( P ) = 0 = ⇒ g | X = 0 . Example: Let ∗ be the generic point of X in scheme theoretic sense . Then g ( ∗ ) =

P ROOFS BY EXAMPLE Algebraic setting (first attempt): Let X = V ( f 1 , . . . , f m ) ⊆ ❈ n be algebraic variety, dim X = d . Let g ( x 1 , . . . , x n ) be polynomial. Call P ∈ X “sufficiently generic” for g if g ( P ) = 0 = ⇒ g | X = 0 . Example: Let ∗ be the generic point of X in scheme theoretic sense . Then g ( ∗ ) = g mod I ( X ) .

P ROOFS BY EXAMPLE Algebraic setting (first attempt): Let X = V ( f 1 , . . . , f m ) ⊆ ❈ n be algebraic variety, dim X = d . Let g ( x 1 , . . . , x n ) be polynomial. Call P ∈ X “sufficiently generic” for g if g ( P ) = 0 = ⇒ g | X = 0 . Example: Let ∗ be the generic point of X in scheme theoretic sense . Then g ( ∗ ) = g mod I ( X ) . Thus g ( ∗ ) = 0 iff g | X = 0.

P ROOFS BY EXAMPLE Algebraic setting (first attempt): Let X = V ( f 1 , . . . , f m ) ⊆ ❈ n be algebraic variety, dim X = d . Let g ( x 1 , . . . , x n ) be polynomial. Call P ∈ X “sufficiently generic” for g if g ( P ) = 0 = ⇒ g | X = 0 . Example: Let ∗ be the generic point of X in scheme theoretic sense . Then g ( ∗ ) = g mod I ( X ) . Thus g ( ∗ ) = 0 iff g | X = 0. � Trivial!

P ROOFS BY EXAMPLE Algebraic setting (first attempt): Let X = V ( f 1 , . . . , f m ) ⊆ ❈ n be algebraic variety, dim X = d . Let g ( x 1 , . . . , x n ) be polynomial. Call P ∈ X “sufficiently generic” for g if g ( P ) = 0 = ⇒ g | X = 0 . Example: Let ∗ be the generic point of X in scheme theoretic sense . Then g ( ∗ ) = g mod I ( X ) . Thus g ( ∗ ) = 0 iff g | X = 0. � Trivial! � Useless.. .

❈ P ROOFS BY EXAMPLE Algebraic setting (first attempt): Let X = V ( f 1 , . . . , f m ) ⊆ ❈ n be algebraic variety, dim X = d . Let g ( x 1 , . . . , x n ) be polynomial. Call P ∈ X “sufficiently generic” for g if g ( P ) = 0 = ⇒ g | X = 0 . Case X = ❈ n . Want P such that g ( P ) = 0 = ⇒ g = 0.

P ROOFS BY EXAMPLE Algebraic setting (first attempt): Let X = V ( f 1 , . . . , f m ) ⊆ ❈ n be algebraic variety, dim X = d . Let g ( x 1 , . . . , x n ) be polynomial. Call P ∈ X “sufficiently generic” for g if g ( P ) = 0 = ⇒ g | X = 0 . Case X = ❈ n . Want P such that g ( P ) = 0 = ⇒ g = 0. Schwartz-Zippel lemma (1979–80; Ore 1922): If A ⊂ ❈ finite, p 1 , . . . , p n independent and uniformly at random from A , then ≤ deg g � � g � = 0 = ⇒ P g ( p 1 , . . . , p n ) = 0 | A | .

❈ P ROOFS BY EXAMPLE Algebraic setting (first attempt): Let X = V ( f 1 , . . . , f m ) ⊆ ❈ n be algebraic variety, dim X = d . Let g ( x 1 , . . . , x n ) be polynomial. Call P ∈ X “sufficiently generic” for g if g ( P ) = 0 = ⇒ g | X = 0 . Case X = ❈ n . Want P such that g ( P ) = 0 = ⇒ g = 0.

P ROOFS BY EXAMPLE Algebraic setting (first attempt): Let X = V ( f 1 , . . . , f m ) ⊆ ❈ n be algebraic variety, dim X = d . Let g ( x 1 , . . . , x n ) be polynomial. Call P ∈ X “sufficiently generic” for g if g ( P ) = 0 = ⇒ g | X = 0 . Case X = ❈ n . Want P such that g ( P ) = 0 = ⇒ g = 0. Combinatorial Nullstellensatz (Alon 1999, weak): If A ⊂ ❈ , | A | > deg g , then g ( A × . . . × A ) = 0 = ⇒ g = 0 .

P ROOFS BY EXAMPLE Algebraic setting (first attempt): Let X = V ( f 1 , . . . , f m ) ⊆ ❈ n be algebraic variety, dim X = d . Let g ( x 1 , . . . , x n ) be polynomial. Call P ∈ X “sufficiently generic” for g if g ( P ) = 0 = ⇒ g | X = 0 . Case X = ❈ . Want P such that g ( P ) = 0 = ⇒ g = 0.

P ROOFS BY EXAMPLE Algebraic setting (first attempt): Let X = V ( f 1 , . . . , f m ) ⊆ ❈ n be algebraic variety, dim X = d . Let g ( x 1 , . . . , x n ) be polynomial. Call P ∈ X “sufficiently generic” for g if g ( P ) = 0 = ⇒ g | X = 0 . Case X = ❈ . Want P such that g ( P ) = 0 = ⇒ g = 0. Lagrange’s theorem (1798): If g ( t ) = a 0 + a 1 t + . . . + a n − 1 t n − 1 + t n , then 1 , � | a i | � � | x | > max = ⇒ g ( x ) � = 0 .

P ROOFS BY EXAMPLE Want: ◮ sufficiently generic example P , ◮ example P easy to construct, ◮ g ( P ) easy to compute, ◮ allow for numerical margin of error.

P ROOFS BY EXAMPLE Main theorem (over ◗ with standard | . | (2019)). Let n irreducible, dim X = d , ◮ X = V ( f 1 , . . . , f m ) ⊆ ◗ ◮ g polynomial, ◮ H := “arithmetic complexity” of ( f 1 , . . . , f m , g ) , ◮ P = ( p 1 , . . . , p n ) ∈ ◗ n such that 0 ≪ H h ( p 1 ) ≪ H h ( p 2 ) ≪ H . . . ≪ H h ( p d ) . Let ε := ε ( H , h ( p d )) . Then � | f i ( P ) | ≤ ε ∀ i and � if = ⇒ g | X = 0 . | g ( P ) | ≤ ε

P ROOFS BY EXAMPLE Remarks ◮ “Robust one-point Nullstellensatz” ◮ Based on ◮ arithmetic B´ ezout theorem [Bost–Gillet–Soul´ e (1991,94), Philippon] ◮ arithmetic Nullstellensatz [Krick–Pardo–Sombra] ◮ new effective Łojasiewicz inequality ◮ Way to remove irreducibility assumption on X . ◮ Way to remove knowledge of dimension of X . ◮ Motivates other “robust Nullstellens¨ atze”. ◮ Motivates more general combinatorial Nullstellens¨ atze.