The Quantitative Combinatorial Nullstellensatz and Integer-Valued Polynomials

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The Quantitative Combinatorial Nullstellensatz and Integer-Valued Polynomials

Uwe Schauz [email protected]

Xi’an Jiaotong-Liverpool University, Suzhou, China

Lille, June 24, 2013

Uwe Schauz (XJTLU) Combinatorial Nullstellensatz Lille, June 24, 2013 1 / 20

(2)

Quantitative Combinatorial Nullstellensatz

Theorem (Quantitative Combinatorial Nullstellensatz)

Let I ₁ , I ₂ , . . . , I _n be finite subsets of a field F , d _j := |I _j | − 1 , and let P ∈ F [X 1 , X 2 , . . . , X n ] .

If deg(P ) ≤ |I ₁ | + |I ₂ | + · · · + |I _n | − n = d ₁ + d ₂ + · · · + d _n then for the coefficient P d

1

,d

2

,...,d

n

of the monomial X ₁ ^d

¹

X ₂ ^d

²

· · · X _n ^d

ⁿ

in P the

following holds:

P d

1

,d

2

,...,d

n

= X

x∈I

₁

×I

₂

×···×I

n

M (x)P (x) ,

with a certain map M : I ₁ × I ₂ × · · · × I _n −→ F not depending on P .

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Quantitative Combinatorial Nullstellensatz Important Corollaries

Theorem (Quantitative Combinatorial Nullstellensatz)

Let I := I 1 × · · · × I n ⊆ F ⁿ and set d j := |I _j | − 1 , d := (d j ) . For polynomials P = P

δ∈ N

ⁿ

P _δ X ^δ ∈ F [X ₁ , . . . , X _n ] of total degree deg(P ) ≤ Σ _j d _j ,

P _d = X

x∈I

N(x) ⁻¹ P (x) , where N (x 1 , . . . , x n ) := Q

j

Q

ξ∈I

_j

\x

j

(x j − ξ) 6= 0 . Corollaries

(i) P d 6= 0 = ⇒

{ x ∈ I ¦ P(x) 6= 0 }

6= 0 . (ii) deg(P ) < Σ j d j = ⇒ P d = 0 = ⇒

{ x ∈ I ¦ P(x) 6= 0 }

6= 1 .

(iii) If F := F q and P 1 , . . . , P m ∈ F [X 1 , . . . , X n ] then X

i

deg(P _i ) < Σ _j d _j

q − 1 = ⇒

x ∈ I ¦ P ₁ (x) = · · · = P _m (x) = 0 6= 1 .

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Quantitative Combinatorial Nullstellensatz Important Corollaries

Theorem (Quantitative Combinatorial Nullstellensatz)

Let I := I 1 × · · · × I n ⊆ F ⁿ and set d j := |I _j | − 1 , d := (d j ) . For polynomials P = P

δ∈ N

ⁿ

P _δ X ^δ ∈ F [X ₁ , . . . , X _n ] of total degree deg(P ) ≤ Σ _j d _j ,

P _d = X

x∈I

N(x) ⁻¹ P (x) , where N (x 1 , . . . , x n ) := Q

j

Q

ξ∈I

_j

\x

j

(x j − ξ) 6= 0 . Corollaries

(i) P d 6= 0 = ⇒

{ x ∈ I ¦ P(x) 6= 0 }

6= 0 . (ii) deg(P ) < Σ j d j = ⇒ P d = 0 = ⇒

{ x ∈ I ¦ P(x) 6= 0 }

6= 1 . (iii) If F := F q and P 1 , . . . , P m ∈ F [X 1 , . . . , X n ] then

X

i

deg(P _i ) < Σ _j d _j

q − 1 = ⇒

x ∈ I ¦ P ₁ (x) = · · · = P _m (x) = 0 6= 1 .

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Quantitative Combinatorial Nullstellensatz Interpolation and Inversion Formulas for the Proof

Theorem (Interpolation Formula)

Let I ⊆ F ⁿ be a d -grid over a field F and y : I −→ F a map.

There exists a unique polynomial P ∈ F [X 1 , . . . , X n ] with partial degrees deg _j (P ) ≤ d _j that interpolates y , i.e., P | _I = y .

The coefficients P _δ of P are given by

P _δ = X

x∈I

M _δ (x) y(x) ,

with certain maps M δ : I −→ F . Corollary (Inversion Formula)

Polynomials P ∈ F [X ₁ , . . . , X _n ] with partial degrees deg _j (P ) ≤ d _j are uniquely determined by P| _I . The coefficients P δ of P are given by

P _δ = X

x∈I

M _δ (x) P (x) .

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Theorem (Interpolation Formula)

Let I ⊆ F ⁿ be a d -grid over a field F and y : I −→ F a map.

There exists a unique polynomial P ∈ F [X 1 , . . . , X n ] with partial degrees deg _j (P ) ≤ d _j that interpolates y , i.e., P | _I = y .

The coefficients P _δ of P are given by

P _δ = X

x∈I

M _δ (x) y(x) ,

with certain maps M δ : I −→ F . Corollary (Inversion Formula)

Polynomials P ∈ F [X ₁ , . . . , X _n ] with partial degrees deg _j (P ) ≤ d _j are uniquely determined by P| _I . The coefficients P δ of P are given by

P _δ = X

x∈I

M _δ (x) P (x) .

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Theorem (Interpolation Formula)

Let I ⊆ F ⁿ be a d -grid over a field F and y : I −→ F a map.

There exists a unique polynomial P ∈ F [X 1 , . . . , X n ] with partial degrees deg _j (P ) ≤ d _j that interpolates y , i.e., P | _I = y .

The coefficients P _δ of P are given by P _δ = X

x∈I

M _δ (x) y(x) ,

with certain maps M δ : I −→ F . Corollary (Inversion Formula)

Polynomials P ∈ F [X ₁ , . . . , X _n ] with partial degrees deg _j (P ) ≤ d _j are uniquely determined by P| _I . The coefficients P δ of P are given by

P _δ = X

x∈I

M _δ (x) P (x) .

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Theorem (Interpolation Formula)

Let I ⊆ F ⁿ be a d -grid over a field F and y : I −→ F a map.

There exists a unique polynomial P ∈ F [X 1 , . . . , X n ] with partial degrees deg _j (P ) ≤ d _j that interpolates y , i.e., P | _I = y .

The coefficients P _δ of P are given by P _δ = X

x∈I

M _δ (x) y(x) ,

with certain maps M δ : I −→ F . Corollary (Inversion Formula)

Polynomials P ∈ F [X ₁ , . . . , X _n ] with partial degrees deg _j (P ) ≤ d _j are uniquely determined by P| _I . The coefficients P δ of P are given by

P _δ = X

x∈I

M _δ (x) P (x) .

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Quantitative Combinatorial Nullstellensatz The Proof

Proof of the Quantitative Combinatorial Nullstellensatz.

Transform P into a trimmed polynomial P /I with (i) (P /I )(x) = P (x) for all x ∈ I ,

(ii) (P /I ) d = P d ,

(iii) deg _j (P /I ) ≤ d j for j = 1, . . . , n . Then

P _d ⁽ⁱⁱ = (P/I) ⁾ _d ⁽ⁱⁱⁱ = ⁾ P

x∈I M _d (x) (P /I)(x) ⁽ⁱ⁾ = P

x∈I M _d (x) P (x) .

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Proof of the Quantitative Combinatorial Nullstellensatz.

Transform P into a trimmed polynomial P /I with (i) (P /I )(x) = P (x) for all x ∈ I ,

(ii) (P /I ) d = P d ,

(iii) deg _j (P /I ) ≤ d j for j = 1, . . . , n . Then

P _d ⁽ⁱⁱ = (P/I) ⁾ _d ⁽ⁱⁱⁱ = ⁾ P

x∈I M _d (x) (P /I)(x) ⁽ⁱ⁾ = P

x∈I M _d (x) P (x) .

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Proof of the Quantitative Combinatorial Nullstellensatz.

Transform P into a trimmed polynomial P /I with (i) (P /I )(x) = P (x) for all x ∈ I ,

(ii) (P /I ) d = P d ,

(iii) deg _j (P /I ) ≤ d j for j = 1, . . . , n . Then

P _d ⁽ⁱⁱ = (P/I) ⁾ _d ⁽ⁱⁱⁱ = ⁾ P

x∈I M _d (x) (P /I)(x) ⁽ⁱ⁾ = P

x∈I M _d (x) P (x) .

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Proof of the Quantitative Combinatorial Nullstellensatz.

Transform P into a trimmed polynomial P /I with (i) (P /I )(x) = P (x) for all x ∈ I ,

(ii) (P /I ) d = P d ,

(iii) deg _j (P /I ) ≤ d j for j = 1, . . . , n . Then

P _d ⁽ⁱⁱ = (P/I) ⁾ _d ⁽ⁱⁱⁱ = ⁾ P

x∈I M _d (x) (P /I)(x) ⁽ⁱ⁾ = P

x∈I M _d (x) P (x) .

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The transformation P 7−→7−→ · · · 7−→ P /X :

deg

₂

6 5 4 d

2

= 3 2 1 0

deg

₁

6 5 4 d

1

2 1 0

Start with P and add successively . . .

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The transformation P 7−→7−→ · · · 7−→ P /X :

deg

₂

6 5 4 d

2

= 3 2 1 0

deg

₁

6 5 4 d

1

2 1 0

|

−− −− −− | | c X ⁽ 0 , 2 ⁾ Y

ξ∈X

₂

(X 2 − ξ) ≡ 0 on X

(15)

The transformation P 7−→7−→ · · · 7−→ P /X :

deg

₂

6 5 4 d

2

= 3 2 1 0

deg

₁

6 5 4 d

1

2 1 0 a a

... a

|

−− −− −− | | c X ⁽ 0 , 2 ⁾ Y

ξ∈X

₂

(X 2 − ξ) ≡ 0 on X

(16)

The transformation P 7−→7−→ · · · 7−→ P /X :

deg

₂

6 5 4 d

2

= 3 2 1 0

deg

₁

6 5 4 d

1

2 1 0

|

−− −− −− | | c X ⁽ 0 , 1 ⁾ Y

ξ∈X

₂

(X 2 − ξ) ≡ 0 on X

(17)

The transformation P 7−→7−→ · · · 7−→ P /X :

deg

₂

6 5 4 d

2

= 3 2 1 0

deg

₁

6 5 4 d

1

2 1 0

|

−− −− −− | | c X ⁽ 0 , 1 ⁾ Y

ξ∈X

₂

(X 2 − ξ) ≡ 0 on X

(18)

The transformation P 7−→7−→ · · · 7−→ P /X :

deg

₂

6 5 4 d

2

= 3 2 1 0

deg

₁

6 5 4 d

1

2 1 0 a a

... a

|

−− −− −− | | c X ⁽ 0 , 1 ⁾ Y

ξ∈X

₂

(X 2 − ξ) ≡ 0 on X

(19)

The transformation P 7−→7−→ · · · 7−→ P /X :

deg

₂

6 5 4 d

2

= 3 2 1 0

deg

₁

6 5 4 d

1

2 1 0 a a

... a

|

−− −− −− | | c X ⁽ 0 , 0 ⁾ Y

ξ∈X

₂

(X 2 − ξ) ≡ 0 on X

(20)

The transformation P 7−→7−→ · · · 7−→ P /X :

deg

₂

6 5 4 d

2

= 3 2 1 0

deg

₁

6 5 4 d

1

2 1 0

a a

... a

|

−− −− −− | | c X ⁽ 1 , 1 ⁾ Y

ξ∈X

₂

(X 2 − ξ) ≡ 0 on X

(21)

The transformation P 7−→7−→ · · · 7−→ P /X :

deg

₂

6 5 4 d

2

= 3 2 1 0

deg

₁

6 5 4 d

1

2 1 0

a a

... a

|

−− −− −− | | c X ⁽ 1 , 0 ⁾ Y

ξ∈X

₂

(X 2 − ξ) ≡ 0 on X

(22)

The transformation P 7−→7−→ · · · 7−→ P /X :

deg

₂

6 5 4 d

2

= 3 2 1 0

deg

₁

6 5 4 d

1

2 1 0

a a

... a

|

−− −− −− | | c X ⁽ 2 , 0 ⁾ Y

ξ∈X

₂

(X 2 − ξ) ≡ 0 on X

(23)

The transformation P 7−→7−→ · · · 7−→ P /X :

deg

₂

6 5 4 d

2

= 3 2 1 0

deg

₁

6 5 4 d

1

2 1 0

a a

... a

|

−− −− −− | | c X ⁽ 2 , 0 ⁾ Y

ξ∈X

₁

(X 1 − ξ) ≡ 0 on X

(24)

The transformation P 7−→7−→ · · · 7−→ P /X :

deg

₂

6 5 4 d

2

= 3 2 1 0

deg

₁

6 5 4 d

1

2 1 0

a a a

...

|

−− −− −− | | c X ⁽ 1 , 0 ⁾ Y

ξ∈X

₁

(X 1 − ξ) ≡ 0 on X

(25)

The transformation P 7−→7−→ · · · 7−→ P /X :

deg

₂

6 5 4 d

2

= 3 2 1 0

deg

₁

6 5 4 d

1

2 1 0

a a a

...

|

−− −− −− | | c X ⁽ 0 , 0 ⁾ Y

ξ∈X

₁

(X 1 − ξ) ≡ 0 on X

(26)

The transformation P 7−→7−→ · · · 7−→ P /X :

deg

₂

6 5 4 d

2

= 3 2 1 0

deg

₁

6 5 4 d

1

2 1 0

a a a

...

|

−− −− −− | | c X ⁽ 1 , 1 ⁾ Y

ξ∈X

₁

(X 1 − ξ) ≡ 0 on X

(27)

The transformation P 7−→7−→ · · · 7−→ P /X :

deg

₂

6 5 4 d

2

= 3 2 1 0

deg

₁

6 5 4 d

1

2 1 0

a a a

...

|

−− −− −− | | c X ⁽ 0 , 1 ⁾ Y

ξ∈X

₁

(X 1 − ξ) ≡ 0 on X

(28)

The transformation P 7−→7−→ · · · 7−→ P /X :

deg

₂

6 5 4 d

2

= 3 2 1 0

deg

₁

6 5 4 d

1

2 1 0

a a a

...

|

−− −− −− | | c X ⁽ 0 , 2 ⁾ Y

ξ∈X

₁

(X 1 − ξ) ≡ 0 on X

(29)

The transformation P 7−→7−→ · · · 7−→ P /X :

deg

₂

6 5 4 d

2

= 3 2 1 0

deg

₁

6 5 4 d

1

2 1 0

`

... ` unchanged!

The trimmed polynomial P /X .

(30)

Quantitative Combinatorial Nullstellensatz Specializations

Specializations of the Quantitative Combinatorial Nullstellensatz

If deg(P ) ≤ Σd , then P d = X

x∈I

N(x) ⁻¹ P (x) Let A ∈ F ^m×n , b ∈ F ^m .

If m ≤ Σd , then per _d (A) = X

x∈I

N(x) ⁻¹

Matrix Poly.

z Q }| { (Ax − b)

Let d v denote the indegree of the vertices v ∈ V of G ~ = (V , ~ E ) and let I _v ⊆ F be a “ list of d _v + 1 colors” so that the set I := Q

v∈V I _v of potential list colorings of G ~ is a d -grid for d := (d _v ) v∈V , then

± |EE| ∓ |EO|

| {z }

Eulerian Subgraphs

= per _d (A( G ~ )

| {z }

Incidence Matrix

) = X

x∈I

N(x) ⁻¹

Graph Poly.

z }| {

Q

−

→ s t∈ E ~

(x _t − x _s )

If ~ L is the arbitrarily oriented line graph of a planar k-regular graph G and d e = k − 1 for all e ∈ E ( G) = V ( ~ L) , then

const · per _d (A( ~ L)) = “ the number of edge k-colorings of G ”

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Non-uniqueness Theorem An Application

Theorem (Alon, Friedland, Kalai)

Every loopless 4-regular multigraph plus one edge G = (V , E ] {e ₀ } ) contains a nontrivial 3-regular subgraph.

(32)

Theorem (Alon, Friedland, Kalai) Every

extended 4-regular graph

z }| {

loopless 4-regular multigraph plus one edge G = (V , E ¯ z }| { E ] {e ₀ }) contains a nontrivial 3-regular subgraph.

(33)

Theorem (Alon, Friedland, Kalai) Every

extended 4-regular graph

z }| {

loopless 4-regular multigraph plus one edge G = (V , E ¯ z }| { E ] {e ₀ }) contains a nontrivial 3-regular subgraph.

A 4-regular graph without 3-regular subgraph

(34)

Theorem (Alon, Friedland, Kalai) Every

extended 4-regular graph

z }| {

loopless 4-regular multigraph plus one edge G = (V , E ¯ z }| { E ] {e ₀ }) contains a nontrivial 3-regular subgraph.

Extended graph

(35)

Theorem (Alon, Friedland, Kalai) Every

extended 4-regular graph

z }| {

loopless 4-regular multigraph plus one edge G = (V , E ¯ z }| { E ] {e ₀ }) contains a nontrivial 3-regular subgraph.

3-regular subgraph

(36)

Theorem (Alon, Friedland, Kalai) Every

extended 4-regular graph

z }| {

loopless 4-regular multigraph plus one edge G = (V , E ¯ z }| { E ] {e ₀ }) contains a nontrivial 3-regular subgraph.

Brikman graph

(37)

Theorem (Alon, Friedland, Kalai) Every

extended 4-regular graph

z }| {

loopless 4-regular multigraph plus one edge G = (V , E ¯ z }| { E ] {e ₀ }) contains a nontrivial 3-regular subgraph.

Subgraph

(38)

Theorem (Alon, Friedland, Kalai) Every

extended 4-regular graph

z }| {

loopless 4-regular multigraph plus one edge G = (V , E ¯ z }| { E ] {e ₀ }) contains a nontrivial 3-regular subgraph.

An other 4-regular graph

(39)

Theorem (Alon, Friedland, Kalai) Every

extended 4-regular graph

z }| {

loopless 4-regular multigraph plus one edge G = (V , E ¯ z }| { E ] {e ₀ }) contains a nontrivial 3-regular subgraph.

I am sure there is one!

(40)

Theorem (Alon, Friedland, Kalai) Every

extended 4-regular graph

z }| {

loopless 4-regular multigraph plus one edge G = (V , E ¯ z }| { E ] {e ₀ }) contains a nontrivial 3-regular subgraph.

Proof.

The subgraphs H ⊆ E ¯ correspond to the points x = (x e ) _e∈ E ¯ of the Boolean grid I := {0, 1} ^E ^¯ ⊆ F 3 ^E ^¯ . E ^¯ ←→ {1,...,n}

V ←→ {1,...,m}

Algebraic Solution:

P v := X

e3v

X e ∈ F 3 [ X e ¦ e ∈ E ¯ ] for all v ∈ V .

Degree Restriction: (3 − 1) P

v

deg(P

v

) = 2|V | = |E| < | E| ¯ = P

e

d

e

.

not exactly one solution exactly one trivial solution

at least one nontrivial solution!

not exactly one solution exactly one trivial solution

at least one nontrivial solution!

not exactly one solution exactly one trivial solution

at least one nontrivial solution!

(41)

Theorem (Alon, Friedland, Kalai) Every

extended 4-regular graph

z }| {

loopless 4-regular multigraph plus one edge G = (V , E ¯ z }| { E ] {e ₀ }) contains a nontrivial 3-regular subgraph.

Proof.

The subgraphs H ⊆ E ¯ correspond to the points x = (x e ) _e∈ E ¯ of the Boolean grid I := {0, 1} ^E ^¯ ⊆ F 3 ^E ^¯ . E ^¯ ←→ {1,...,n}

V ←→ {1,...,m}

Algebraic Solution:

P v := X

e3v

X e ∈ F 3 [ X e ¦ e ∈ E ¯ ] for all v ∈ V .

Degree Restriction: (3 − 1) P

v

deg(P

v

) = 2|V | = |E| < | E| ¯ = P

e

d

e

.

not exactly one solution exactly one trivial solution

at least one nontrivial solution!

not exactly one solution exactly one trivial solution

at least one nontrivial solution!

not exactly one solution exactly one trivial solution

at least one nontrivial solution!

(42)

Theorem (Alon, Friedland, Kalai) Every

extended 4-regular graph

z }| {

loopless 4-regular multigraph plus one edge G = (V , E ¯ z }| { E ] {e ₀ }) contains a nontrivial 3-regular subgraph.

Proof.

The subgraphs H ⊆ E ¯ correspond to the points x = (x e ) _e∈ E ¯ of the Boolean grid I := {0, 1} ^E ^¯ ⊆ F 3 ^E ^¯ . E ^¯ ←→ {1,...,n}

V ←→ {1,...,m}

Algebraic Solution:

P v := X

e3v

X e ∈ F 3 [ X e ¦ e ∈ E ¯ ] for all v ∈ V .

Degree Restriction: (3 − 1) P

v

deg(P

v

) = 2|V | = |E| < | E| ¯ = P

e

d

e

.

not exactly one solution exactly one trivial solution

at least one nontrivial solution!

not exactly one solution exactly one trivial solution

at least one nontrivial solution!

not exactly one solution exactly one trivial solution

at least one nontrivial solution!

(43)

Theorem (Alon, Friedland, Kalai) Every

extended 4-regular graph

z }| {

loopless 4-regular multigraph plus one edge G = (V , E ¯ z }| { E ] {e ₀ }) contains a nontrivial 3-regular subgraph.

Proof.

The subgraphs H ⊆ E ¯ correspond to the points x = (x e ) _e∈ E ¯ of the Boolean grid I := {0, 1} ^E ^¯ ⊆ F 3 ^E ^¯ . E ^¯ ←→ {1,...,n}

V ←→ {1,...,m}

Algebraic Solution:

P v := X

e3v

X e ∈ F 3 [ X e ¦ e ∈ E ¯ ] for all v ∈ V .

Degree Restriction: (3 − 1) P

v

deg(P

v

) = 2|V | = |E| < | E| ¯ = P

e

d

e

.

not exactly one solution exactly one trivial solution

at least one nontrivial solution!

not exactly one solution exactly one trivial solution

at least one nontrivial solution!

not exactly one solution exactly one trivial solution

at least one nontrivial solution!

(44)

Non-uniqueness Theorem Prime-Characteristic Problem

Theorem (Non-uniqueness Theorem)

Let R be a commutative ring with 1 6= 0 and P ∈ R[X 1 , X 2 , . . . , X n ] . If deg(P ) < n then

x ∈ {0, 1} ⁿ ¦ P (x) 6= 0 6= 1 .

Theorem (Version of Warning’s Theorem)

Let F q be the field of order q and P ₁ , . . . , P _m ∈ F q [X ₁ , X ₂ , . . . , X _n ] . If (q − 1) P m

i=1 deg(P i ) < n then

x ∈ {0, 1} ⁿ ¦ P 1 (x) = · · · = P m (x) = 0 6= 1 .

Proof.

Define P := Q m

i=1 (1 − P _i ^q−1 ) and apply the Non-uniqueness Theorem.

(45)

Theorem (Non-uniqueness Theorem)

Let R be a commutative ring with 1 6= 0 and P ∈ R[X 1 , X 2 , . . . , X n ] . If deg(P ) < n then

x ∈ {0, 1} ⁿ ¦ P (x) 6= 0 6= 1 .

Theorem (Version of Warning’s Theorem)

Let F q be the field of order q and P ₁ , . . . , P _m ∈ F q [X ₁ , X ₂ , . . . , X _n ] . If (q − 1) P m

i=1 deg(P i ) < n then

x ∈ {0, 1} ⁿ ¦ P 1 (x) = · · · = P m (x) = 0 6= 1 . Proof.

Define P := Q m

i=1 (1 − P _i ^q−1 ) and apply the Non-uniqueness Theorem.

(46)

Theorem (Non-uniqueness Theorem)

Let R be a commutative ring with 1 6= 0 and P ∈ R[X 1 , X 2 , . . . , X n ] . If deg(P ) < n then

x ∈ {0, 1} ⁿ ¦ P (x) 6= 0 6= 1 .

Theorem (Version of Warning’s Theorem)

Let F q be the field of order q and P ₁ , . . . , P _m ∈ F q [X ₁ , X ₂ , . . . , X _n ] . If (q − 1) P m

i=1 deg(P i ) < n then

x ∈ {0, 1} ⁿ ¦ P 1 (x) = · · · = P m (x) = 0 6= 1 . Proof.

Define P := Q m

i=1 (1 − P _i ^q−1 ) and apply the Non-uniqueness Theorem.

(47)

Theorem (Non-existence of Lagrange Functions)

Let P ∈ Z k [X ₁ , . . . , X _n ] . If k is not prime, and (k, n) 6= (4, 1) , then { x ∈ Z k n ¦ P(x) 6= 0 }

6= 1 .

(48)

Interpolation-Valued Polynomials Periodicity

− ¹ ₈ X ⁴ + ³ ₄ X ³ − ⁷ ₈ X ² − ³ ₄ X + 1 : Z −−−→ Q

= −3 ^X ₄ + ^X ₂

− ^X ₁ + ^X ₀

: Z −− −−−→ Z

−−→ →

Z 9 3-periodic:

−3 ^X ₄ + ^X ₂

− ^X ₁ + ^X ₀

: Z 3 −−−→ Z 9 P(x+3)=P(x)

f : Z −→ Z 9 is 3-periodic ⇐⇒ f : Z 3 −→ Z 9

Z ^Z ₉

³

⊆ Z ^Z ₉

⁶

⊆ Z ^Z 9

(49)

Interpolation-Valued Polynomials Periodicity

− ¹ ₈ X ⁴ + ³ ₄ X ³ − ⁷ ₈ X ² − ³ ₄ X + 1 : Z −−−→ Q

= −3 ^X ₄ + ^X ₂

− ^X ₁ + ^X ₀

: Z −−−→ Z

− ¯ 3 ^X ₄

+ ¯ 1 ^X ₂

− ¯ 1 ^X ₁

+ ¯ 1 ^X ₀

: Z −−−→ Z 9 3-periodic

− ¯ 3 ^X ₄

+ ¯ 1 ^X ₂

− ¯ 1 ^X ₁

+ ¯ 1 ^X ₀

| {z }

∈ Z

9

X Z

3

⊆ Z

9

X Z

: Z 3 −−−→ Z 9

| {z }

∈ Z

^Z39

⊆ Z

^Z9

¯

a := a + 9Z

Z 9 X Z

3

, → Z ^Z ₉

³

is injective.

Z 9 X Z

3

= {f ∈ Z ^Z ₉

³

¦ f arises from a ˆ f ∈ Q[X ] }

(50)

Interpolation-Valued Polynomials Primepower Characteristic

Good news:

Z p

^β

X Z p

^α

= ( Z p

^β

) ^Z

^p^α

for primes p .

Any function Z p

^α

−→ Z p

^β

arises from a polynomial, in particular the Lagrange Function L , which maps nonzeros to zero and zero to one.

If β = 1 then L : Z p

^α

−→ Z p has degree p ^α − 1 .

Bad news:

Z p

^β

X Z q

^α

= ( Z p

^β

) ^Z

^p⁰

= Z p

^β

for different primes p and q .

Only constant functions Z q

^α

−→ Z p

^β

arise from a polynomial.

The general case Z r

X Z s

is a certain mixture of these cases.

(51)

Good news:

Z p

^β

X Z p

^α

= ( Z p

^β

) ^Z

^p^α

for primes p .

Any function Z p

^α

−→ Z p

^β

arises from a polynomial, in particular the Lagrange Function L , which maps nonzeros to zero and zero to one.

If β = 1 then L : Z p

^α

−→ Z p has degree p ^α − 1 .

Bad news:

Z p

^β

X Z q

^α

= ( Z p

^β

) ^Z

^p⁰

= Z p

^β

for different primes p and q .

Only constant functions Z q

^α

−→ Z p

^β

arise from a polynomial.

The general case Z r

X Z s

is a certain mixture of these cases.

(52)

Good news:

Z p

^β

X Z p

^α

= ( Z p

^β

) ^Z

^p^α

for primes p .

Any function Z p

^α

−→ Z p

^β

arises from a polynomial, in particular the Lagrange Function L , which maps nonzeros to zero and zero to one.

If β = 1 then L : Z p

^α

−→ Z p has degree p ^α − 1 .

Bad news:

Z p

^β

X Z q

^α

= ( Z p

^β

) ^Z

^p⁰

= Z p

^β

for different primes p and q .

Only constant functions Z q

^α

−→ Z p

^β

arise from a polynomial.

The general case Z r

X Z s

is a certain mixture of these cases.

(53)

Good news:

Z p

^β

X Z p

^α

= ( Z p

^β

) ^Z

^p^α

for primes p .

Any function Z p

^α

−→ Z p

^β

arises from a polynomial, in particular the Lagrange Function L , which maps nonzeros to zero and zero to one.

If β = 1 then L : Z p

^α

−→ Z p has degree p ^α − 1 .

Bad news:

Z p

^β

X Z q

^α

= ( Z p

^β

) ^Z

^p⁰

= Z p

^β

for different primes p and q . Only constant functions Z q

^α

−→ Z p

^β

arise from a polynomial.

The general case Z r

X Z s

is a certain mixture of these cases.

(54)

Good news:

Z p

^β

X Z p

^α

= ( Z p

^β

) ^Z

^p^α

for primes p .

Any function Z p

^α

−→ Z p

^β

arises from a polynomial, in particular the Lagrange Function L , which maps nonzeros to zero and zero to one.

If β = 1 then L : Z p

^α

−→ Z p has degree p ^α − 1 .

Bad news:

Z p

^β

X Z q

^α

= ( Z p

^β

) ^Z

^p⁰

= Z p

^β

for different primes p and q . Only constant functions Z q

^α

−→ Z p

^β

arise from a polynomial.

The general case Z r

X Z s

is a certain mixture of these cases.

(55)

Interpolation-Valued Polynomials Generalized Combinatorial Nullstellensatz

Theorem (Quantitative Combinatorial Nullstellensatz)

Let I ₁ , I ₂ , . . . , I _n be finite subsets of a field F , d _j := |I _j | − 1 , d := (d _j ) . For polynomials P = P

δ∈ N

ⁿ

P _δ X ^δ ∈ F [X ₁ , . . . , X _n ] with deg(P) ≤ Σ _j d _j ,

P _d = X

x∈I

1

×I

2

×···×I

n

M (x)P (x) , with certain M(x) ∈ F \0 .

Theorem (Generalized Quantitative Combinatorial Nullstellensatz) For j = 1, . . . , n let I j ⊆ Z be such that the distances x 2 − x 1 between any two elements of I _j is not a multiple of p , d _j := |I _j | − 1 , d := (d _j ) . For integer valued polynomials P = P

δ∈ N

ⁿ

P _δ X ^δ ∈ Q [X ₁ , . . . , X _n ] with deg(P ) ≤ Σ j d j , which we view as map Z ⁿ −→ Z p ,

((p − 1)!) ⁿ P d

| {z }

∈ Z

+pZ = X

x∈I

₁

×I

₂

×···×I

n

M (x)P(x) , with certain M (x) ∈ Z p \0 .

(56)

Theorem (Quantitative Combinatorial Nullstellensatz)

Let I ₁ , I ₂ , . . . , I _n be finite subsets of a field F , d _j := |I _j | − 1 , d := (d _j ) . For polynomials P = P

δ∈ N

ⁿ

P _δ X ^δ ∈ F [X ₁ , . . . , X _n ] with deg(P) ≤ Σ _j d _j ,

P _d = X

x∈I

1

×I

2

×···×I

n

M (x)P (x) , with certain M(x) ∈ F \0 .

Theorem (Generalized Quantitative Combinatorial Nullstellensatz) For j = 1, . . . , n let I j ⊆ Z be such that the distances x 2 − x 1 between any two elements of I _j is not a multiple of p , d _j := |I _j | − 1 , d := (d _j ) . For integer valued polynomials P = P

δ∈ N

ⁿ

P _δ X ^δ ∈ Q [X ₁ , . . . , X _n ] with deg(P ) ≤ Σ j d j , which we view as map Z ⁿ −→ Z p ,

((p − 1)!) ⁿ P d

| {z }

∈ Z

+pZ = X

x∈I

₁

×I

₂

×···×I

n

M (x)P(x) , with certain M (x) ∈ Z p \0 .

(57)

Theorem (Quantitative Combinatorial Nullstellensatz)

Let I ₁ , I ₂ , . . . , I _n be finite subsets of a field F , d _j := |I _j | − 1 , d := (d _j ) . For polynomials P = P

δ∈ N

ⁿ

P _δ X ^δ ∈ F [X ₁ , . . . , X _n ] with deg(P) ≤ Σ _j d _j ,

P _d = X

x∈I

1

×I

2

×···×I

n

M (x)P (x) , with certain M(x) ∈ F \0 .

Theorem (Generalized Quantitative Combinatorial Nullstellensatz) For j = 1, . . . , n let I j ⊆ Z be such that the distances x 2 − x 1 between any two elements of I _j is not a multiple of p , d _j := |I _j | − 1 , d := (d _j ) . For integer valued polynomials P = P

δ∈ N

ⁿ

P _δ X ^δ ∈ Q [X ₁ , . . . , X _n ] with deg(P ) ≤ Σ j d j , which we view as map Z ⁿ −→ Z p ,

((p − 1)!) ⁿ P d

| {z }

∈ Z

+pZ = X

x∈I

₁

×I

₂

×···×I

n

M (x)P(x) , with certain M (x) ∈ Z p \0 .

(58)

Interpolation-Valued Polynomials Generalized Corollaries

Theorem (Non-uniqueness Theorem)

Let I ₁ , I ₂ , . . . , I _n be finite subsets of a field F and P ∈ F [X ₁ , X ₂ , . . . , X _n ] . If deg(P ) < |I ₁ | + |I ₂ | + · · · + |I _n | − n then

{ x ∈ I ₁ × I ₂ × · · · × I _n ¦ P(x) 6= 0 } 6= 1 .

Theorem (Generalized Non-uniqueness Theorem)

For j = 1, . . . , n let I _j ⊆ Z be such that the distances x ₂ − x ₁ between any two elements of I j is not a multiple of p . Let P ∈ Q[X 1 , . . . , X n ] be integer valued. We view view P as map Z ⁿ −→ Z p .

If deg(P ) < |I ₁ | + |I ₂ | + · · · + |I _n | − n then

{ x ∈ I ₁ × I ₂ × · · · × I _n ¦ P(x) 6= 0 } 6= 1 .

(59)

Theorem (Non-uniqueness Theorem)

Let I ₁ , I ₂ , . . . , I _n be finite subsets of a field F and P ∈ F [X ₁ , X ₂ , . . . , X _n ] . If deg(P ) < |I ₁ | + |I ₂ | + · · · + |I _n | − n then

{ x ∈ I ₁ × I ₂ × · · · × I _n ¦ P(x) 6= 0 } 6= 1 .

Theorem (Generalized Non-uniqueness Theorem)

For j = 1, . . . , n let I _j ⊆ Z be such that the distances x ₂ − x ₁ between any two elements of I j is not a multiple of p . Let P ∈ Q[X 1 , . . . , X n ] be integer valued. We view view P as map Z ⁿ −→ Z p .

If deg(P ) < |I ₁ | + |I ₂ | + · · · + |I _n | − n then

{ x ∈ I ₁ × I ₂ × · · · × I _n ¦ P(x) 6= 0 } 6= 1 .

(60)

Theorem (Non-uniqueness Theorem)

Let I ₁ , I ₂ , . . . , I _n be finite subsets of a field F and P ∈ F [X ₁ , X ₂ , . . . , X _n ] . If deg(P ) < |I ₁ | + |I ₂ | + · · · + |I _n | − n then

{ x ∈ I ₁ × I ₂ × · · · × I _n ¦ P(x) 6= 0 } 6= 1 .

Theorem (Generalized Non-uniqueness Theorem)

For j = 1, . . . , n let I _j ⊆ Z be such that the distances x ₂ − x ₁ between any two elements of I j is not a multiple of p . Let P ∈ Q[X 1 , . . . , X n ] be integer valued. We view view P as map Z ⁿ −→ Z p .

If deg(P ) < |I ₁ | + |I ₂ | + · · · + |I _n | − n then

{ x ∈ I ₁ × I ₂ × · · · × I _n ¦ P(x) 6= 0 } 6= 1 .

(61)

Theorem (Version of Warning’s Theorem)

Let I ₁ , I ₂ , . . . , I _n be finite subsets of a the finite field F q , and let P 1 , P 2 , . . . , P m ∈ F q [X 1 , X 2 , . . . , X n ] .

If (q − 1) P m

i=1 deg(P i ) < |I ₁ | + |I ₂ | + · · · + |I _n | − n then

x ∈ I 1 × I 2 × · · · × I n ¦ P 1 (x) = · · · = P m (x) = 0 6= 1 .

Theorem (Generalized Version of Warning’s Theorem)

For j = 1, . . . , n let I _j ⊆ Z be such that the distances x ₂ − x ₁ between any two elements of I j is not a multiple of a given prime p , and let P ₁ , . . . , P _m ∈ Z [X ₁ , . . . , X _n ] and 0 < k ₁ , . . . , k _m ∈ Z .

If P m

i=1 (p ^k

ⁱ

− 1) deg(P i ) < |I ₁ | + |I ₂ | + · · · + |I _n | − n then

x ∈ X ¦ ∀ i : p ^k

ⁱ

P _i (x)

6= 1 .

(62)

Theorem (Version of Warning’s Theorem)

Let I ₁ , I ₂ , . . . , I _n be finite subsets of a the finite field F q , and let P 1 , P 2 , . . . , P m ∈ F q [X 1 , X 2 , . . . , X n ] .

If (q − 1) P m

i=1 deg(P i ) < |I ₁ | + |I ₂ | + · · · + |I _n | − n then

x ∈ I 1 × I 2 × · · · × I n ¦ P 1 (x) = · · · = P m (x) = 0 6= 1 .

Theorem (Generalized Version of Warning’s Theorem)

For j = 1, . . . , n let I _j ⊆ Z be such that the distances x ₂ − x ₁ between any two elements of I j is not a multiple of a given prime p , and let P ₁ , . . . , P _m ∈ Z [X ₁ , . . . , X _n ] and 0 < k ₁ , . . . , k _m ∈ Z .

If P m

i=1 (p ^k

ⁱ

− 1) deg(P i ) < |I ₁ | + |I ₂ | + · · · + |I _n | − n then

x ∈ X ¦ ∀ i : p ^k

ⁱ

P _i (x)

6= 1 .

(63)

Theorem (Olson’s Theorem for primes p )

Any sequence of (p − 1)m + 1 elements of Z ^m p contains a subsequence that sums to zero.

Theorem (Generalized Olson Theorem) Any sequence of 1 + P

i (p ^k

ⁱ

− 1) elements of Z _p

^k1

× Z _p

^k2

× · · · × Z p

^km

contains a subsequence that sums to zero.

Conjecture (Alon, Friedland, Kalai)

For any integer k ≥ 2 , any sequence of (k − 1)n + 1 elements of Z ⁿ _k contains a subsequence that sums to zero.

(64)

Theorem (Olson’s Theorem for primes p )

Any sequence of (p − 1)m + 1 elements of Z ^m p contains a subsequence that sums to zero.

Theorem (Generalized Olson Theorem) Any sequence of 1 + P

i (p ^k

ⁱ

− 1) elements of Z _p

^k1

× Z _p

^k2

× · · · × Z p

^km

contains a subsequence that sums to zero.

Conjecture (Alon, Friedland, Kalai)

For any integer k ≥ 2 , any sequence of (k − 1)n + 1 elements of Z ⁿ _k contains a subsequence that sums to zero.

(65)

Theorem (Olson’s Theorem for primes p )

Any sequence of (p − 1)m + 1 elements of Z ^m p contains a subsequence that sums to zero.

Theorem (Generalized Olson Theorem) Any sequence of 1 + P

i (p ^k

ⁱ

− 1) elements of Z _p

^k1

× Z _p

^k2

× · · · × Z p

^km

contains a subsequence that sums to zero.

Conjecture (Alon, Friedland, Kalai)

For any integer k ≥ 2 , any sequence of (k − 1)n + 1 elements of Z ⁿ _k contains a subsequence that sums to zero.

(66)

Theorem (Alon, Friedland, Kalai)

For prime powers q holds the following: Every loopless (2q−2)-regular multigraph plus one edge contains a nontrivial q-regular subgraph.

Conjecture (Alon, Friedland, Kalai).

This holds for any integer q ≥ 2 .

(67)

Theorem (Alon, Friedland, Kalai)

For prime powers q holds the following: Every loopless (2q−2)-regular multigraph plus one edge contains a nontrivial q-regular subgraph.

Conjecture (Alon, Friedland, Kalai).

This holds for any integer q ≥ 2 .

(68)

Interpolation-Valued Polynomials Non-Primepower Characteristic

Assume

f ∈ Z 12

X Z 50

.

Then

φ◦f ◦ϑ ⁻¹ ∈ ( Z 2

²

× Z 3

¹

× Z 5

⁰

)

X 1 , X 2 , X 3

Z 2

¹

× Z 3

⁰

× Z 5

²

= Z ^Z ₄

²

× Z ^Z ₃

¹

× Z ^Z ₁

²⁵

∼ = Z ² 4 × Z 3 , with the Chinese Reminder Isomorphisms

φ: Z 12 −→ Z 2

²

× Z 3

¹

× Z 5

⁰

and ϑ : Z 50 −→ Z 2

¹

× Z 3

⁰

× Z 5

²

.

Hence,

φ ◦ f ◦ ϑ ⁻¹ = (g , c , 0) ∈ Z ^Z ₄

²

× Z ^{0} ₃ × {0} ^Z

²⁵

. It follows that

f = 9g + 4c = −3g + 4c ,

with a 2-periodic g : Z −→ {0, 1, 2, 3} ⊆ Z 12 and a constant c ∈ {0, 1, 2} ⊆ Z 12 . In other words,

f : Z −→ Z 12 is 2-periodic and f (1) − f (0) ∈ {0, 3, 6, 9} ⊆ Z 12 .

(69)

Interpolation-Valued Polynomials Goodbye

The Quantitative Combinatorial Nullstellensatz and Integer-Valued Polynomials