applications of the combinatorial nullstellensatz in ...applications of the combinatorial...
TRANSCRIPT
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Applications of the Combinatorial Nullstellensatz in Additive Combinatorics
Applications of the Combinatorial Nullstellensatzin Additive Combinatorics
Eric Balandraud
Additive Combinatorics in BordeauxLundi 11 Avril 2016
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Applications of the Combinatorial Nullstellensatz in Additive Combinatorics
The Combinatorial Nullstellensatz
The Combinatorial Nullstellensatz
Theorem (Alon, 1999)
K a field and P a polynomial K[X1, . . . ,Xd ].
If deg(P) =∑d
i=1 ki and P has a non zero coefficient for∏d
i=1 Xkii ,
then whatever A1, . . . , Ad , subsets of K such that |Ai | > ki , thereexists (a1, . . . , ad) ∈ A1 × · · · × Ad so that:
P(a1, . . . , ad) 6= 0.
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Applications of the Combinatorial Nullstellensatz in Additive Combinatorics
The Combinatorial Nullstellensatz
The Combinatorial Nullstellensatz
Theorem (Alon, 1999)
K a field and P a polynomial K[X1, . . . ,Xd ].If deg(P) =
∑di=1 ki and P has a non zero coefficient for
∏di=1 X
kii ,
then whatever A1, . . . , Ad , subsets of K such that |Ai | > ki , thereexists (a1, . . . , ad) ∈ A1 × · · · × Ad so that:
P(a1, . . . , ad) 6= 0.
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Applications of the Combinatorial Nullstellensatz in Additive Combinatorics
The Combinatorial Nullstellensatz
The Combinatorial Nullstellensatz
Theorem (Alon, 1999)
K a field and P a polynomial K[X1, . . . ,Xd ].If deg(P) =
∑di=1 ki and P has a non zero coefficient for
∏di=1 X
kii ,
then whatever A1, . . . , Ad , subsets of K such that |Ai | > ki , thereexists (a1, . . . , ad) ∈ A1 × · · · × Ad so that:
P(a1, . . . , ad) 6= 0.
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Applications of the Combinatorial Nullstellensatz in Additive Combinatorics
The Combinatorial Nullstellensatz
Another formulation
Theorem (Alon, 1999)
K a field and P a polynomial K[X1, . . . ,Xd ]. Let A1, . . . , Ad
subsets of K. Setting gi (Xi ) =∏
ai∈Ai(Xi − ai ). If P vanishes on
A1 × · · · × Ad , there exist hi ∈ K[X1, . . . ,Xd ], withdeg(hi ) 6 deg(P)− deg(gi ) such that:
P =d∑
i=1
higi .
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Applications of the Combinatorial Nullstellensatz in Additive Combinatorics
The Combinatorial Nullstellensatz
The polynomial method
CombinatorialProblem
(P, (A1, . . . ,Ad))−−−−−−−−−−−−→
CalculusProblem
Solution orContradiction
← Non zero Coefficient
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Applications of the Combinatorial Nullstellensatz in Additive Combinatorics
The Combinatorial Nullstellensatz
The polynomial method
CombinatorialProblem
(P, (A1, . . . ,Ad))−−−−−−−−−−−−→
CalculusProblem
Solution orContradiction
← Non zero Coefficient
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Applications of the Combinatorial Nullstellensatz in Additive Combinatorics
Addition Theorems in Fp
Addition Theorems in Fp
I Cauchy-Davenport
I Dias da Silva-Hamidoune (Erdos-Heilbronn)
I Set of Subsums
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Applications of the Combinatorial Nullstellensatz in Additive Combinatorics
Addition Theorems in FpCauchy-Davenport
Cauchy-Davenport
Theorem (Cauchy-Davenport - 1813, 1935)
p a prime number, A and B two subsets of Fp, then:
|A + B| > min {p, |A|+ |B| − 1} .
(p > (|A| − 1) + |B − 1|)A× B, ∏
c∈A+B
(X + Y − c)
The coefficient of X |A|−1Y |B|−1 is((|A|−1)+(|B|−1)
|A|−1
)6= 0.
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Applications of the Combinatorial Nullstellensatz in Additive Combinatorics
Addition Theorems in FpCauchy-Davenport
Cauchy-Davenport
Theorem (Cauchy-Davenport - 1813, 1935)
p a prime number, A and B two subsets of Fp, then:
|A + B| > min {p, |A|+ |B| − 1} .
(p > (|A| − 1) + |B − 1|)A× B,
∏c∈A+B
(X + Y − c)
The coefficient of X |A|−1Y |B|−1 is((|A|−1)+(|B|−1)
|A|−1
)6= 0.
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Applications of the Combinatorial Nullstellensatz in Additive Combinatorics
Addition Theorems in FpCauchy-Davenport
Cauchy-Davenport
Theorem (Cauchy-Davenport - 1813, 1935)
p a prime number, A and B two subsets of Fp, then:
|A + B| > min {p, |A|+ |B| − 1} .
(p > (|A| − 1) + |B − 1|)A× B, ∏
c∈A+B
(X + Y − c)
The coefficient of X |A|−1Y |B|−1 is((|A|−1)+(|B|−1)
|A|−1
)6= 0.
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Applications of the Combinatorial Nullstellensatz in Additive Combinatorics
Addition Theorems in FpCauchy-Davenport
Cauchy-Davenport
Theorem (Cauchy-Davenport - 1813, 1935)
p a prime number, A and B two subsets of Fp, then:
|A + B| > min {p, |A|+ |B| − 1} .
(p > (|A| − 1) + |B − 1|)A× B, ∏
c∈A+B
(X + Y − c)
The coefficient of X |A|−1Y |B|−1 is((|A|−1)+(|B|−1)
|A|−1
)6= 0.
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Applications of the Combinatorial Nullstellensatz in Additive Combinatorics
Addition Theorems in FpDias da Silva-Hamidoune
Dias da Silva-Hamidoune
Conjecture (Erdos-Heilbronn - 1964)
p a prime number, A ⊂ Fp, then:
|{a1 + a2 | ai ∈ A, a1 6= a2}| > min{p, 2|A| − 3}
Define:h∧A = {a1 + · · ·+ ah | ai ∈ A, ai 6= aj}
Theorem (Dias da Silva, Hamidoune - 1994)
Let p be a prime number and A ⊂ Fp. Let h ∈ [1, |A|] , one has,
|h∧A| > min{p, 1 + h(|A| − h)}.
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Applications of the Combinatorial Nullstellensatz in Additive Combinatorics
Addition Theorems in FpDias da Silva-Hamidoune
Dias da Silva-Hamidoune
Conjecture (Erdos-Heilbronn - 1964)
p a prime number, A ⊂ Fp, then:
|{a1 + a2 | ai ∈ A, a1 6= a2}| > min{p, 2|A| − 3}
Define:h∧A = {a1 + · · ·+ ah | ai ∈ A, ai 6= aj}
Theorem (Dias da Silva, Hamidoune - 1994)
Let p be a prime number and A ⊂ Fp. Let h ∈ [1, |A|] , one has,
|h∧A| > min{p, 1 + h(|A| − h)}.
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Applications of the Combinatorial Nullstellensatz in Additive Combinatorics
Addition Theorems in FpDias da Silva-Hamidoune
Dias da Silva-Hamidoune
Conjecture (Erdos-Heilbronn - 1964)
p a prime number, A ⊂ Fp, then:
|{a1 + a2 | ai ∈ A, a1 6= a2}| > min{p, 2|A| − 3}
Define:h∧A = {a1 + · · ·+ ah | ai ∈ A, ai 6= aj}
Theorem (Dias da Silva, Hamidoune - 1994)
Let p be a prime number and A ⊂ Fp. Let h ∈ [1, |A|] , one has,
|h∧A| > min{p, 1 + h(|A| − h)}.
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Applications of the Combinatorial Nullstellensatz in Additive Combinatorics
Addition Theorems in FpDias da Silva-Hamidoune
(p > h(|A| − h))
A1 ={a1, . . . , a|A|−h, a|A|−h+1
}...
.... . .
Ah−1 ={a1, . . . , a|A|−1
}Ah =
{a1, . . . , a|A|−1, a|A|
},
∏c∈h∧A
(X1 + · · ·+ Xh − c)
∏16i<j6h
(Xj − Xi )
.
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Applications of the Combinatorial Nullstellensatz in Additive Combinatorics
Addition Theorems in FpDias da Silva-Hamidoune
(p > h(|A| − h))
A1 ={a1, . . . , a|A|−h, a|A|−h+1
}...
.... . .
Ah−1 ={a1, . . . , a|A|−1
}Ah =
{a1, . . . , a|A|−1, a|A|
},
∏c∈h∧A
(X1 + · · ·+ Xh − c)
∏16i<j6h
(Xj − Xi )
.
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Applications of the Combinatorial Nullstellensatz in Additive Combinatorics
Addition Theorems in FpSet of Subsums
Set of Subsums
Let A be a subset of Fp, define
Σ(A) =
{∑x∈I
x | ∅ ⊂ I ⊂ A
}
Theorem (B. - 2012)
p a odd prime number, A ⊂ Fp, such that A ∩ (−A) = ∅. One has
|Σ(A)| > min
{p, 1 +
|A|(|A|+ 1)
2
},
|Σ∗(A)| > min
{p,
|A|(|A|+ 1)
2
}.
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Applications of the Combinatorial Nullstellensatz in Additive Combinatorics
Addition Theorems in FpSet of Subsums
Set of Subsums
Let A be a subset of Fp, define
Σ∗(A) =
{∑x∈I
x | ∅ ( I ⊂ A
}
Theorem (B. - 2012)
p a odd prime number, A ⊂ Fp, such that A ∩ (−A) = ∅. One has
|Σ(A)| > min
{p, 1 +
|A|(|A|+ 1)
2
},
|Σ∗(A)| > min
{p,
|A|(|A|+ 1)
2
}.
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Applications of the Combinatorial Nullstellensatz in Additive Combinatorics
Addition Theorems in FpSet of Subsums
Set of Subsums
Let A be a subset of Fp, define
Σ∗(A) =
{∑x∈I
x | ∅ ( I ⊂ A
}
Theorem (B. - 2012)
p a odd prime number, A ⊂ Fp, such that A ∩ (−A) = ∅. One has
|Σ(A)| > min
{p, 1 +
|A|(|A|+ 1)
2
},
|Σ∗(A)| > min
{p,
|A|(|A|+ 1)
2
}.
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Applications of the Combinatorial Nullstellensatz in Additive Combinatorics
Addition Theorems in FpSet of Subsums
Set of Subsums
Let A be a subset of Fp, define
Σ∗(A) =
{∑x∈I
x | ∅ ( I ⊂ A
}
Theorem (B. - 2012)
p a odd prime number, A ⊂ Fp, such that A ∩ (−A) = ∅. One has
|Σ(A)| > min
{p, 1 +
|A|(|A|+ 1)
2
},
|Σ∗(A)| > min
{p,
|A|(|A|+ 1)
2
}.
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Applications of the Combinatorial Nullstellensatz in Additive Combinatorics
Addition Theorems in FpSet of Subsums
(p > d(d+1)2 + 1), A = {2a1, . . . , 2ad}:
Σ(A) =
(d∑
i=1
ai
)+
d∑i=1
{−ai , ai}
A1 = {a1, . . . , ad ,−a1}...
. . .
Ad = {a1, . . . , ad ,−a1, . . . ,−ad} ,
∏c∈Σ(A)
(X1 + · · ·+ Xd − c)
∏16i<j6d
(X 2j − X 2
i
) .
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Applications of the Combinatorial Nullstellensatz in Additive Combinatorics
Addition Theorems in FpSet of Subsums
(p > d(d+1)2 + 1), A = {2a1, . . . , 2ad}:
Σ(A) =
(d∑
i=1
ai
)+
d∑i=1
{−ai , ai}
A1 = {a1, . . . , ad ,−a1}...
. . .
Ad = {a1, . . . , ad ,−a1, . . . ,−ad} ,
∏c∈Σ(A)
(X1 + · · ·+ Xd − c)
∏16i<j6d
(X 2j − X 2
i
) .
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Applications of the Combinatorial Nullstellensatz in Additive Combinatorics
Addition Theorems in FpSet of Subsums
(p > d(d+1)2 + 1), A = {2a1, . . . , 2ad}:
Σ(A) =
(d∑
i=1
ai
)+
d∑i=1
{−ai , ai}
A1 = {a1, . . . , ad ,−a1}...
. . .
Ad = {a1, . . . , ad ,−a1, . . . ,−ad} ,
∏c∈Σ(A)
(X1 + · · ·+ Xd − c)
∏16i<j6d
(X 2j − X 2
i
) .
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Applications of the Combinatorial Nullstellensatz in Additive Combinatorics
Addition Theorems in FpSet of Subsums
Binomial Determinants
(a1, . . . , adb1, . . . , bd
)=
∣∣∣∣∣∣∣∣∣
(a1b1
) (a1b2
). . .
(a1bd
)(a2b1
) (a2b2
). . .
(a2bd
)...
......(ad
b1
) (adb2
). . .
( adbd )
)∣∣∣∣∣∣∣∣∣ .
Dn,h =
(n − h, n − h + 1, . . . , n − 1
0, 1, . . . , (h − 1)
)
= 1 6= 0,
Dd =
(d , d + 1, . . . , 2d − 10, 2, . . . , 2(d − 1)
)
= 2d(d−1)/2 6= 0
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Applications of the Combinatorial Nullstellensatz in Additive Combinatorics
Addition Theorems in FpSet of Subsums
Binomial Determinants
(a1, . . . , adb1, . . . , bd
)=
∣∣∣∣∣∣∣∣∣
(a1b1
) (a1b2
). . .
(a1bd
)(a2b1
) (a2b2
). . .
(a2bd
)...
......(ad
b1
) (adb2
). . .
( adbd )
)∣∣∣∣∣∣∣∣∣ .
Dn,h =
(n − h, n − h + 1, . . . , n − 1
0, 1, . . . , (h − 1)
)
= 1 6= 0,
Dd =
(d , d + 1, . . . , 2d − 10, 2, . . . , 2(d − 1)
)
= 2d(d−1)/2 6= 0
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Applications of the Combinatorial Nullstellensatz in Additive Combinatorics
Addition Theorems in FpSet of Subsums
Binomial Determinants
(a1, . . . , adb1, . . . , bd
)=
∣∣∣∣∣∣∣∣∣
(a1b1
) (a1b2
). . .
(a1bd
)(a2b1
) (a2b2
). . .
(a2bd
)...
......(ad
b1
) (adb2
). . .
( adbd )
)∣∣∣∣∣∣∣∣∣ .
Dn,h =
(n − h, n − h + 1, . . . , n − 1
0, 1, . . . , (h − 1)
)= 1 6= 0,
Dd =
(d , d + 1, . . . , 2d − 10, 2, . . . , 2(d − 1)
)= 2d(d−1)/2 6= 0
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Applications of the Combinatorial Nullstellensatz in Additive Combinatorics
Additive results on sequences
Additive results on sequences
I Erdos-Ginzburg-Ziv
I Snevily’s conjecture (Arsovsky)
I Kemnitz’ conjecture (Reiher)
I Problem “a la Vinatier”
I Nullstellensatz for sequences
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Applications of the Combinatorial Nullstellensatz in Additive Combinatorics
Additive results on sequences
Erdos-Ginzburg-Ziv
The permanent Lemma
Theorem (Alon - 1999)
K a field, A an n × n matrix with non zero permanent, b ∈ Kn,and Si ⊂ K, i = 1..n, |Si | = 2. There existss = (s1, . . . , sn) ∈ S1 × · · · × Sn, such that As and b arecoordinatewise distincts.
∏ni=1 Ai =
∏ni=1 Si ,
n∏i=1
n∑j=1
ai ,jXj − bi
,
coefficient of∏n
i=1 Xi is Per(A) 6= 0.
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Applications of the Combinatorial Nullstellensatz in Additive Combinatorics
Additive results on sequences
Erdos-Ginzburg-Ziv
The permanent Lemma
Theorem (Alon - 1999)
K a field, A an n × n matrix with non zero permanent, b ∈ Kn,and Si ⊂ K, i = 1..n, |Si | = 2. There existss = (s1, . . . , sn) ∈ S1 × · · · × Sn, such that As and b arecoordinatewise distincts.
∏ni=1 Ai =
∏ni=1 Si ,
n∏i=1
n∑j=1
ai ,jXj − bi
,
coefficient of∏n
i=1 Xi is Per(A) 6= 0.
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Applications of the Combinatorial Nullstellensatz in Additive Combinatorics
Additive results on sequences
Erdos-Ginzburg-Ziv
Erdos-Ginzburg-Ziv
Theorem (Erdos, Ginzburg, Ziv - 1961)
G abelian finite group, |G | = n. Whathever (g1, g2, . . . , g2n−1)elements of G . There exists a zerosum subsequence of length n.
∏p−1i=1 Ai =
∏p−1i=1 {gi , gi+p−1}
A =
1 . . . 1...
...1 . . . 1
︸ ︷︷ ︸
p−1
,
b = (−g2p−1 + 1, . . . ,−g2p−1 + (p − 1)).
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Applications of the Combinatorial Nullstellensatz in Additive Combinatorics
Additive results on sequences
Erdos-Ginzburg-Ziv
Erdos-Ginzburg-Ziv
Theorem (Erdos, Ginzburg, Ziv - 1961)
G abelian finite group, |G | = n. Whathever (g1, g2, . . . , g2n−1)elements of G . There exists a zerosum subsequence of length n.
∏p−1i=1 Ai =
∏p−1i=1 {gi , gi+p−1}
A =
1 . . . 1...
...1 . . . 1
︸ ︷︷ ︸
p−1
,
b = (−g2p−1 + 1, . . . ,−g2p−1 + (p − 1)).
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Applications of the Combinatorial Nullstellensatz in Additive Combinatorics
Additive results on sequences
Snevily’s conjecture
Snevily’s Conjecture
G a finite abelian group of odd order.a1, . . . , ak , distinct elements
There is π, such that
b1, . . . , bk , distinct elements
a1 + bπ(1), . . . , ak + bπ(k)
are pairwise distincts
.
G = Z/nZ (Dasgupta, Karolyi, Serra, Szegedy - 2001),∏ki=1 Ai = {gai | i = 1..k}k ⊂ Fk
2d,
P(X1, . . . ,Xk) =∏
16j<i6k
(Xi − Xj) (αiXi − αjXj) ,
with αi = gbi .
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Applications of the Combinatorial Nullstellensatz in Additive Combinatorics
Additive results on sequences
Snevily’s conjecture
Snevily’s Conjecture
G a finite abelian group of odd order.a1, . . . , ak , distinct elements There is π, such thatb1, . . . , bk , distinct elements a1 + bπ(1), . . . , ak + bπ(k)
are pairwise distincts.
G = Z/nZ (Dasgupta, Karolyi, Serra, Szegedy - 2001),∏ki=1 Ai = {gai | i = 1..k}k ⊂ Fk
2d,
P(X1, . . . ,Xk) =∏
16j<i6k
(Xi − Xj) (αiXi − αjXj) ,
with αi = gbi .
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Applications of the Combinatorial Nullstellensatz in Additive Combinatorics
Additive results on sequences
Snevily’s conjecture
Snevily’s Conjecture
G a finite abelian group of odd order.a1, . . . , ak , distinct elements There is π, such thatb1, . . . , bk , distinct elements a1 + bπ(1), . . . , ak + bπ(k)
are pairwise distincts.
G = Z/nZ (Dasgupta, Karolyi, Serra, Szegedy - 2001),∏ki=1 Ai = {gai | i = 1..k}k ⊂ Fk
2d,
P(X1, . . . ,Xk) =∏
16j<i6k
(Xi − Xj) (αiXi − αjXj) ,
with αi = gbi .
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Applications of the Combinatorial Nullstellensatz in Additive Combinatorics
Additive results on sequences
Kemnitz conjecture
Kemnitz conjecture
Theorem (Ronyai - 2000)
In a sequence of 4p − 2 elements ((ai , bi )) of F2p, there is a 0-sum
subsequence of length p:
∏4p−2i=1 Ai = {0, 1}4p−2,
1−
(4p−2∑i=1
aiXi
)p−11−
(4p−2∑i=1
biXi
)p−1
×
1−
(4p−2∑i=1
Xi
)p−1 ∑
I∈[1,4p−2]|I |=p
∏i∈I
Xi − 2
+ (2L0(X )).
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Applications of the Combinatorial Nullstellensatz in Additive Combinatorics
Additive results on sequences
Kemnitz conjecture
Kemnitz conjecture
Theorem (Ronyai - 2000)
In a sequence of 4p − 2 elements ((ai , bi )) of F2p, there is a 0-sum
subsequence of length p:
∏4p−2i=1 Ai = {0, 1}4p−2,
1−
(4p−2∑i=1
aiXi
)p−11−
(4p−2∑i=1
biXi
)p−1
×
1−
(4p−2∑i=1
Xi
)p−1 ∑
I∈[1,4p−2]|I |=p
∏i∈I
Xi − 2
+ (2L0(X )).
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Applications of the Combinatorial Nullstellensatz in Additive Combinatorics
Additive results on sequences
Problem “a la Vinatier”
Problem “a la Vinatier”
Theorem (Gacs, Heger, Nagy, Palvogyi - 2010)
In Fnq, n 6 q, Hi ,j = {X |Xi = Xj}, whenever H ⊂
⋃i 6=j Hi ,j .
I H = Hi ,j ,
I n = q, H = {X |α(Xi − Xj) +∑
Xk = 0},I n = q − 1, H = {X |Xj +
∑Xk = 0}.
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Applications of the Combinatorial Nullstellensatz in Additive Combinatorics
Additive results on sequences
Problem “a la Vinatier”
Problem “a la Vinatier”
Theorem (Gacs, Heger, Nagy, Palvogyi - 2010)
In Fnq, n 6 q, Hi ,j = {X |Xi = Xj}, whenever H ⊂
⋃i 6=j Hi ,j .
I H = Hi ,j ,
I n = q, H = {X |α(Xi − Xj) +∑
Xk = 0},I n = q − 1, H = {X |Xj +
∑Xk = 0}.
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Applications of the Combinatorial Nullstellensatz in Additive Combinatorics
Additive results on sequences
Problem “a la Vinatier”
Theorem(a1, . . . , aq) ∈ Fq
q
There is no pairwise distinct (b1, . . . , bq) ∈ Fqq such that∑q
i=1 aibi = 0.
⇐⇒ There are (a, b) ∈ Fq, b 6= 0 such thatai = a + b, aj = a− b, and k 6= i , j , ak = a.
∏qi=1 Ai = Fq
q,
G (Y ) =
( k∑i=1
Yi
)q−1
− 1
∣∣∣∣∣∣∣∣∣ak−1
1 ak−11 Y1 . . . Y k−1
1
ak−12 ak−1
2 Y2 . . . Y k−12
......
. . ....
ak−1k ak−1
k Yk . . . Y k−1k
∣∣∣∣∣∣∣∣∣ .
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Applications of the Combinatorial Nullstellensatz in Additive Combinatorics
Additive results on sequences
Problem “a la Vinatier”
Theorem(a1, . . . , aq) ∈ Fq
q
There is no pairwise distinct (b1, . . . , bq) ∈ Fqq such that∑q
i=1 aibi = 0.
⇐⇒ There are (a, b) ∈ Fq, b 6= 0 such thatai = a + b, aj = a− b, and k 6= i , j , ak = a.
∏qi=1 Ai = Fq
q,
G (Y ) =
( k∑i=1
Yi
)q−1
− 1
∣∣∣∣∣∣∣∣∣ak−1
1 ak−11 Y1 . . . Y k−1
1
ak−12 ak−1
2 Y2 . . . Y k−12
......
. . ....
ak−1k ak−1
k Yk . . . Y k−1k
∣∣∣∣∣∣∣∣∣ .
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Applications of the Combinatorial Nullstellensatz in Additive Combinatorics
Additive results on sequences
Nullstellensatz for sequences
A question of Erdos
A = (a1, . . . , a`) a sequence of F×p .SA: set of (0− 1)-solutions of
a1x1 + · · ·+ a`x` = 0.
SA = A⊥ ∩ {0, 1}`
Setdim(A) = dim(< SA >).
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Applications of the Combinatorial Nullstellensatz in Additive Combinatorics
Additive results on sequences
Nullstellensatz for sequences
A question of Erdos
A = (a1, . . . , a`) a sequence of F×p .SA: set of (0− 1)-solutions of
a1x1 + · · ·+ a`x` = 0.
SA = A⊥ ∩ {0, 1}`
Setdim(A) = dim(< SA >).
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Applications of the Combinatorial Nullstellensatz in Additive Combinatorics
Additive results on sequences
Nullstellensatz for sequences
Theorem (B.-Girard, 2014)
A = (a1, . . . , a`) a sequence of ` > p elements of F×p :
dim(A) = `− 1.
Theorem (B.-Girard, - 2014)
A = (a1, . . . , ap) a sequence of p elements of F×p :
I dim(A) = 1,(a1, . . . , ap) = (r , . . . , r).
I dim(A) = p − 2, ∃t ∈ [1, p − 3],
(aσ(1), . . . , aσ(p)) = (r , . . . , r︸ ︷︷ ︸t
,−r , . . . ,−r︸ ︷︷ ︸p−2−t
,−(t+1)r ,−(t+1)r).
I dim(A) = p − 1.
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Applications of the Combinatorial Nullstellensatz in Additive Combinatorics
Additive results on sequences
Nullstellensatz for sequences
Theorem (B.-Girard, 2014)
A = (a1, . . . , a`) a sequence of ` > p elements of F×p :
dim(A) = `− 1.
Theorem (B.-Girard, - 2014)
A = (a1, . . . , ap) a sequence of p elements of F×p :
I dim(A) = 1,(a1, . . . , ap) = (r , . . . , r).
I dim(A) = p − 2, ∃t ∈ [1, p − 3],
(aσ(1), . . . , aσ(p)) = (r , . . . , r︸ ︷︷ ︸t
,−r , . . . ,−r︸ ︷︷ ︸p−2−t
,−(t+1)r ,−(t+1)r).
I dim(A) = p − 1.
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Applications of the Combinatorial Nullstellensatz in Additive Combinatorics
Additive results on sequences
Nullstellensatz for sequences
SA ⊂ SB ,
Σi = Σ(Si ), Si = (aj ∈ A : bj/aj = λi )
∏Ai =
∏Σi ,
P(X1, . . . ,Xd) =
d∑
i=1
λiXi︸ ︷︷ ︸∑i∈I bi
(
()p−1 − 1).
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Applications of the Combinatorial Nullstellensatz in Additive Combinatorics
Additive results on sequences
Nullstellensatz for sequences
SA ⊂ SB ,
Σi = Σ(Si ), Si = (aj ∈ A : bj/aj = λi )
∏Ai =
∏Σi ,
P(X1, . . . ,Xd) =
d∑
i=1
λiXi︸ ︷︷ ︸∑i∈I bi
d∑
i=1
Xi︸ ︷︷ ︸∑i∈I ai
p−1
− 1
.