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

Proofs by example

Benjamin Matschke

Boston University

Number Theory Seminar Harvard, Oct. 2019

PROOFS BY EXAMPLE

Proofs by example

PROOFS BY EXAMPLE

To prove a general statement by verifying it for a single example.

PROOFS BY EXAMPLE

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

PROOFS BY EXAMPLE

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

PROOFS 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

PROOFS 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.

PROOFS 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”:

PROOFS 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.

PROOFS 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, . . .?

PROOFS BY EXAMPLE

Another example: Thales’ theorem Thales of Miletus ∼ 600 BC

PROOFS BY EXAMPLE

Another example: Thales’ theorem Thales of Miletus ∼ 600 BC

PROOFS BY EXAMPLE

Another example: Thales’ theorem Thales of Miletus ∼ 600 BC

PROOFS BY EXAMPLE

Another example: Thales’ theorem Can “Proof by example” work? Thales of Miletus ∼ 600 BC

PROOFS BY EXAMPLE Algebraic setting

❈

PROOFS BY EXAMPLE Algebraic setting (first attempt):

❈

PROOFS BY EXAMPLE Algebraic setting (first attempt):

Let X = V(f1, . . . , fm) ⊆ ❈n be algebraic variety, dim X = d.

PROOFS BY EXAMPLE Algebraic setting (first attempt):

Let X = V(f1, . . . , fm) ⊆ ❈n be algebraic variety, dim X = d. Let g(x1, . . . , xn) be polynomial.

PROOFS BY EXAMPLE Algebraic setting (first attempt):

Let X = V(f1, . . . , fm) ⊆ ❈n be algebraic variety, dim X = d. Let g(x1, . . . , xn) be polynomial. Call P ∈ X “sufficiently generic” for g if g(P) = 0 = ⇒ g|X = 0.

PROOFS BY EXAMPLE Algebraic setting (first attempt):

Let X = V(f1, . . . , fm) ⊆ ❈n be algebraic variety, dim X = d. Let g(x1, . . . , xn) be polynomial. Call P ∈ X “sufficiently generic” for g if g(P) = 0 = ⇒ g|X = 0.

PROOFS BY EXAMPLE Algebraic setting (first attempt):

Let X = V(f1, . . . , fm) ⊆ ❈n be algebraic variety, dim X = d. Let g(x1, . . . , xn) 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.

PROOFS BY EXAMPLE Algebraic setting (first attempt):

Let X = V(f1, . . . , fm) ⊆ ❈n be algebraic variety, dim X = d. Let g(x1, . . . , xn) 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(∗) =

PROOFS BY EXAMPLE Algebraic setting (first attempt):

Let X = V(f1, . . . , fm) ⊆ ❈n be algebraic variety, dim X = d. Let g(x1, . . . , xn) 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).

PROOFS BY EXAMPLE Algebraic setting (first attempt):

Let X = V(f1, . . . , fm) ⊆ ❈n be algebraic variety, dim X = d. Let g(x1, . . . , xn) 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.

PROOFS BY EXAMPLE Algebraic setting (first attempt):

Let X = V(f1, . . . , fm) ⊆ ❈n be algebraic variety, dim X = d. Let g(x1, . . . , xn) 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!

PROOFS BY EXAMPLE Algebraic setting (first attempt):

Let X = V(f1, . . . , fm) ⊆ ❈n be algebraic variety, dim X = d. Let g(x1, . . . , xn) 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.. .

PROOFS BY EXAMPLE Algebraic setting (first attempt):

Let X = V(f1, . . . , fm) ⊆ ❈n be algebraic variety, dim X = d. Let g(x1, . . . , xn) 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. ❈

PROOFS BY EXAMPLE Algebraic setting (first attempt):

Let X = V(f1, . . . , fm) ⊆ ❈n be algebraic variety, dim X = d. Let g(x1, . . . , xn) 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, p1, . . . , pn independent and uniformly at random from A, then g = 0 = ⇒ P

• g(p1, . . . , pn) = 0
• ≤ deg g

|A| .

PROOFS BY EXAMPLE Algebraic setting (first attempt):

Let X = V(f1, . . . , fm) ⊆ ❈n be algebraic variety, dim X = d. Let g(x1, . . . , xn) 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. ❈

PROOFS BY EXAMPLE Algebraic setting (first attempt):

Let X = V(f1, . . . , fm) ⊆ ❈n be algebraic variety, dim X = d. Let g(x1, . . . , xn) 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.

PROOFS BY EXAMPLE Algebraic setting (first attempt):

Let X = V(f1, . . . , fm) ⊆ ❈n be algebraic variety, dim X = d. Let g(x1, . . . , xn) 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.

PROOFS BY EXAMPLE Algebraic setting (first attempt):

Let X = V(f1, . . . , fm) ⊆ ❈n be algebraic variety, dim X = d. Let g(x1, . . . , xn) 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) = a0 + a1t + . . . + an−1tn−1 + tn, then |x| > max

• 1, |ai|
• =

⇒ g(x) = 0.

PROOFS BY EXAMPLE

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

PROOFS BY EXAMPLE

Main theorem (over ◗ with standard | . | (2019)). Let ◮ X = V(f1, . . . , fm) ⊆ ◗

n irreducible, dim X = d,

◮ g polynomial, ◮ H := “arithmetic complexity” of (f1, . . . , fm, g), ◮ P = (p1, . . . , pn) ∈ ◗n such that 0 ≪H h(p1) ≪H h(p2) ≪H . . . ≪H h(pd). Let ε := ε(H, h(pd)). Then if |fi(P)| ≤ ε ∀i and |g(P)| ≤ ε

• =

⇒ g|X = 0.

PROOFS 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.

PROOFS BY EXAMPLE

A comparison: Let X = V(f1, . . . , fm). Hilbert’s Nullstellensatz: g|X = 0 ⇐ ⇒ gN =

i λifi for some N and some polynomials λi

Proof by example scheme: g|X = 0 ⇐ ⇒ g(P) ≈ 0 for some sufficiently generic P close to X

PROOFS BY EXAMPLE

A comparison: Let X = V(f1, . . . , fm). Hilbert’s Nullstellensatz: g|X = 0 ⇐ ⇒ gN =

i λifi for some N and some polynomials λi

Proof by example scheme: g|X = 0 ⇐ ⇒ g(P) ≈ 0 for some sufficiently generic P close to X new witness for g|X = 0.

PROOFS BY EXAMPLE

Example: Thales’ theorem

PROOFS BY EXAMPLE

Example: Thales’ theorem

PROOFS BY EXAMPLE

Example: Thales’ theorem Choose p1 = 0.1234567890123.

PROOFS BY EXAMPLE

Example: Thales’ theorem Choose p1 = 0.1234567890123. Compute p2 =

• 1 − p2

1 up to 1300 digits of precision.

PROOFS BY EXAMPLE

Example: Thales’ theorem Choose p1 = 0.1234567890123. Compute p2 =

• 1 − p2

1 up to 1300 digits of precision.

works!

PROOFS BY EXAMPLE

Measuring dimension by example:

PROOFS BY EXAMPLE

Measuring dimension by example: If ◮ P sufficiently generic and close to X, and ◮ | det([e1, e2, . . . , ed, ∇f1(P), . . . , ∇fn−d(P)])| > ε, then dim X = d.

PROOFS BY EXAMPLE

Measuring dimension by example: If ◮ P sufficiently generic and close to X, and ◮ | det([e1, e2, . . . , ed, ∇f1(P), . . . , ∇fn−d(P)])| > ε, then dim X = d. Note: ε is mild. Equivalence if X is smooth.

PROOFS BY EXAMPLE

Can we decide whether or not g|X = 0?

PROOFS BY EXAMPLE

Can we decide whether or not g|X = 0? – Yes! Dichotomy theorem:

PROOFS BY EXAMPLE

Can we decide whether or not g|X = 0? – Yes! Dichotomy theorem: If P sufficiently generic and close enough to X, then either Case 1: |g(P)| ≤ ε and g|X = 0. Case 2: |g(P)| ≥ 2ε and g|X = 0.

PROOFS BY EXAMPLE

Future topics:

• 1. Better bounds
• 2. Equivalence to arithmetic Nullstellensatz
• 3. Combinatorial Nullstellensatz for varieties

Proofs by examples (e.g. Thales, Pappus, Desargues) Robust combinatorial/probabilistic Nullstellens¨ atze

• 4. Comparison with Gr¨
• bner bases
• 5. Continuation of sequences

Thank you