Additive combinatorics in Fp and the polynomial methodmath.univ-lille1.fr/~bhowmik/seminaire... ·...
Transcript of Additive combinatorics in Fp and the polynomial methodmath.univ-lille1.fr/~bhowmik/seminaire... ·...
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Additive combinatorics in Fp and the polynomial method
Additive combinatorics in Fp
and the polynomial method
Eric Balandraud
Journees estivales de la Methode Polynomiale
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Additive combinatorics in Fp and the polynomial method
The Combinatorial Nullstellensatz
Theorem (Alon)
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 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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Additive combinatorics in Fp and the polynomial method
The Combinatorial Nullstellensatz
Theorem (Alon)
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 ki
i ,
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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Additive combinatorics in Fp and the polynomial method
The Combinatorial Nullstellensatz
Theorem (Alon)
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 ki
i ,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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Additive combinatorics in Fp and the polynomial method
Another formulation
TheoremK 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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Additive combinatorics in Fp and the polynomial method
Addition Theorems
Three Addition Theorems in Fp
I Cauchy-Davenport
I Dias da Silva-Hamidoune (Erdos-Heilbronn)
I Set of Subsums
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Additive combinatorics in Fp and the polynomial method
Addition Theorems
Cauchy-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} .
proof: If A + B ⊂ C , with |C | = min{p − 1, |A|+ |B| − 2}, thepolynomial ∏
c∈C
(X + Y − c)
would vanish on A× B and the coefficient of X |A|−1Y |C |−|A|−1 is( |C ||A|−1
)6= 0.
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Additive combinatorics in Fp and the polynomial method
Addition Theorems
Cauchy-Davenport
Cauchy-Davenport
Theorem (Cauchy-Davenport - 1813, 1935)
p a prime number, A and B two subsets of Fp, then:
|A + B| > min {p, 1 + (|A| − 1) + (|B| − 1)} .
proof: If A + B ⊂ C , with |C | = min{p − 1, |A|+ |B| − 2}, thepolynomial ∏
c∈C
(X + Y − c)
would vanish on A× B and the coefficient of X |A|−1Y |C |−|A|−1 is( |C ||A|−1
)6= 0.
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Additive combinatorics in Fp and the polynomial method
Addition Theorems
Cauchy-Davenport
Cauchy-Davenport
Theorem (Cauchy-Davenport - 1813, 1935)
p a prime number, A and B two subsets of Fp, then:
|A + B| > min {p, 1 + (|A| − 1) + (|B| − 1)} .
proof: If A + B ⊂ C , with |C | = min{p − 1, |A|+ |B| − 2}, thepolynomial ∏
c∈C
(X + Y − c)
would vanish on A× B and the coefficient of X |A|−1Y |C |−|A|−1 is( |C ||A|−1
)6= 0.
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Additive combinatorics in Fp and the polynomial method
Addition Theorems
Dias 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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Additive combinatorics in Fp and the polynomial method
Addition Theorems
Dias 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| − 2) + 1}
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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Additive combinatorics in Fp and the polynomial method
Addition Theorems
Dias 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| − 2) + 1}
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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Additive combinatorics in Fp and the polynomial method
Addition Theorems
Dias 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| − 2) + 1}
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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Additive combinatorics in Fp and the polynomial method
Addition Theorems
Dias da Silva-Hamidoune
Proof: Setting i0 = max{0, h(|A| − h)− p + 1},(h(|A| − h)− p + 1 < h). one considers the polynomial
(X0 + · · ·+ Xh−1)h(|A|−h)−i0
∏06i<j6h−1
(Xj − Xi )
,
it has degree h(|A| − h)− i0 + h(h−1)2 ,
and the sets:
A0 ={
a1, . . . , a|A|−h
}A1 =
{a1, . . . , a|A|−h, a|A|−h+1
}...
.... . .
Ai0−1 ={
a1, . . . , a|A|−h, . . . , a|A|−h+i0−1
}Ai0 =
{a1, . . . , a|A|−h, . . . , a|A|−h+i0−1, a|A|−h+i0 , a|A|−h+i0+1
}...
. . .
Ah−2 ={
a1, . . . , a|A|−1
}Ah−1 =
{a1, . . . , a|A|−1, a|A|
},
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Additive combinatorics in Fp and the polynomial method
Addition Theorems
Dias da Silva-Hamidoune
Proof: Setting i0 = max{0, h(|A| − h)− p + 1},(h(|A| − h)− p + 1 < h). one considers the polynomial
(X0 + · · ·+ Xh−1)h(|A|−h)−i0
∏06i<j6h−1
(Xj − Xi )
,
it has degree h(|A| − h)− i0 + h(h−1)2 , and the sets:
A0 ={
a1, . . . , a|A|−h
}A1 =
{a1, . . . , a|A|−h, a|A|−h+1
}...
.... . .
Ai0−1 ={
a1, . . . , a|A|−h, . . . , a|A|−h+i0−1
}Ai0 =
{a1, . . . , a|A|−h, . . . , a|A|−h+i0−1, a|A|−h+i0 , a|A|−h+i0+1
}...
. . .
Ah−2 ={
a1, . . . , a|A|−1
}Ah−1 =
{a1, . . . , a|A|−1, a|A|
},
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Additive combinatorics in Fp and the polynomial method
Addition Theorems
Set of Subsums
Set of Subsums
Let A be a subset of Fp, define
Σ(A) =
{∑x∈I
x | ∅ ⊂ I ⊂ A
}
Theorem (Olson - 1968)
Let A ⊂ Fp. If A ∩ (−A) = ∅, then
|Σ(A)| > min
{p + 3
2,|A|(|A|+ 1)
2
}.
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Additive combinatorics in Fp and the polynomial method
Addition Theorems
Set of Subsums
Set of Subsums
Let A be a subset of Fp, define
Σ∗(A) =
{∑x∈I
x | ∅ ( I ⊂ A
}
Theorem (Olson - 1968)
Let A ⊂ Fp. If A ∩ (−A) = ∅, then
|Σ(A)| > min
{p + 3
2,|A|(|A|+ 1)
2
}.
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Additive combinatorics in Fp and the polynomial method
Addition Theorems
Set of Subsums
Set of Subsums
Let A be a subset of Fp, define
Σ∗(A) =
{∑x∈I
x | ∅ ( I ⊂ A
}
Theorem (Olson - 1968)
Let A ⊂ Fp. If A ∩ (−A) = ∅, then
|Σ(A)| > min
{p + 3
2,|A|(|A|+ 1)
2
}.
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Additive combinatorics in Fp and the polynomial method
Addition Theorems
Set of Subsums
Theorem (B.)
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
}.
Denote A = {2a1, . . . , 2ad}.
Σ(A) =∑
i∈[1,d ]
{0, 2ai} =
∑i∈[1,d ]
ai
+∑
i∈[1,d ]
{−ai , ai}︸ ︷︷ ︸d terms
.
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Additive combinatorics in Fp and the polynomial method
Addition Theorems
Set of Subsums
Theorem (B.)
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
}.
Denote A = {2a1, . . . , 2ad}.
Σ(A) =∑
i∈[1,d ]
{0, 2ai} =
∑i∈[1,d ]
ai
+∑
i∈[1,d ]
{−ai , ai}︸ ︷︷ ︸d terms
.
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Additive combinatorics in Fp and the polynomial method
Addition Theorems
Set of Subsums
Theorem (B.)
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
}.
Conjecture (Selfridge - 1976)
p a prime number, A a maximal zerosum free subset of Z/pZ ,then:
|A| = max
{k | k(k + 1)
2< p
}.
Denote A = {2a1, . . . , 2ad}.
Σ(A) =∑
i∈[1,d ]
{0, 2ai} =
∑i∈[1,d ]
ai
+∑
i∈[1,d ]
{−ai , ai}︸ ︷︷ ︸d terms
.
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Additive combinatorics in Fp and the polynomial method
Addition Theorems
Set of Subsums
Theorem (B.)
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
}.
Denote A = {2a1, . . . , 2ad}.
Σ(A) =∑
i∈[1,d ]
{0, 2ai} =
∑i∈[1,d ]
ai
+∑
i∈[1,d ]
{−ai , ai}︸ ︷︷ ︸d terms
.
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Additive combinatorics in Fp and the polynomial method
Addition Theorems
Set of Subsums
Theorem (B.)
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
}.
Denote A = {2a1, . . . , 2ad}.
Σ(A) =∑
i∈[1,d ]
{0, 2ai} =
∑i∈[1,d ]
ai
+∑
i∈[1,d ]
{−ai , ai}︸ ︷︷ ︸d terms
.
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Additive combinatorics in Fp and the polynomial method
Addition Theorems
Set of Subsums
Let i0 = max{0, d(d+1)2 − p + 1}, one has
t = d(d+1)2 − i0 = min{d(d+1)
2 , p − 1}
(X0 + · · ·+ Xd−1)t
∏06i<j6d−1
(X 2
j − X 2i
)
A0 = {a1, . . . , ad}A1 = {a1, . . . , ad ,−a1}...
. . .
Ai0−1 = {a1, . . . , ad ,−a1, . . . ,−ai0−1}Ai0 = {a1, . . . , ad ,−a1, . . . ,−ai0−1,−ai0 ,−ai0+1}...
. . .
Ad−1 = {a1, . . . , ad ,−a1, . . . . . . ,−ad} .
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Additive combinatorics in Fp and the polynomial method
Addition Theorems
Set of Subsums
Let i0 = max{0, d(d+1)2 − p + 1}, one has
t = d(d+1)2 − i0 = min{d(d+1)
2 , p − 1}
(X0 + · · ·+ Xd−1)t
∏06i<j6d−1
(X 2
j − X 2i
)A0 = {a1, . . . , ad}A1 = {a1, . . . , ad ,−a1}...
. . .
Ai0−1 = {a1, . . . , ad ,−a1, . . . ,−ai0−1}Ai0 = {a1, . . . , ad ,−a1, . . . ,−ai0−1,−ai0 ,−ai0+1}...
. . .
Ad−1 = {a1, . . . , ad ,−a1, . . . . . . ,−ad} .
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Additive combinatorics in Fp and the polynomial method
Addition Theorems
Set of Subsums
Let i0 = max{0, d(d+1)2 − p + 1}, one has
t = d(d+1)2 − i0 = min{d(d+1)
2 , p − 1}
(X0 + · · ·+ Xd−1)t
∏06i<j6d−1
(X 2
j − X 2i
)A0 = {a1, . . . , ad}A1 = {a1, . . . , ad ,−a1}...
. . .
Ai0−1 = {a1, . . . , ad ,−a1, . . . ,−ai0−1}Ai0 = {a1, . . . , ad ,−a1, . . . ,−ai0−1,−ai0 ,−ai0+1}...
. . .
Ad−1 = {a1, . . . , ad ,−a1, . . . . . . ,−ad} .
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Additive combinatorics in Fp and the polynomial method
Addition Theorems
Set of Subsums
(X0 + · · ·+ Xd−1)t
∏06i<j6d−1
(X 2
j − X 2i
)
=
∑(t0,...,td−1)∑d−1
i=0 ti=t
t!∏d−1i=0 ti !
d−1∏i=0
X tii
1 X 2
0 . . . X2(d−1)0
1 X 21 . . . X
2(d−1)1
......
...
1 X 2d−1 . . . X
2(d−1)d−1
It suffices to prove that the following binomial determinant isnon zero:
Dd ,i0 =
(d − 1, d , . . . , d − 2 + i0, d + i0, . . . , 2d − 1
0, 2, . . . , 2(i0 − 1), 2i0, . . . , 2(d − 1)
).
![Page 28: Additive combinatorics in Fp and the polynomial methodmath.univ-lille1.fr/~bhowmik/seminaire... · Additive combinatorics in Fp and the polynomial method The Combinatorial Nullstellensatz](https://reader034.fdocuments.in/reader034/viewer/2022050111/5f48d3be62c3da5adb64be94/html5/thumbnails/28.jpg)
Additive combinatorics in Fp and the polynomial method
Addition Theorems
Set of Subsums
(X0 + · · ·+ Xd−1)t
∏06i<j6d−1
(X 2
j − X 2i
)
=
∑(t0,...,td−1)∑d−1
i=0 ti=t
t!∏d−1i=0 ti !
d−1∏i=0
X tii
1 X 2
0 . . . X2(d−1)0
1 X 21 . . . X
2(d−1)1
......
...
1 X 2d−1 . . . X
2(d−1)d−1
It suffices to prove that the following binomial determinant isnon zero:
Dd ,i0 =
(d − 1, d , . . . , d − 2 + i0, d + i0, . . . , 2d − 1
0, 2, . . . , 2(i0 − 1), 2i0, . . . , 2(d − 1)
).
![Page 29: Additive combinatorics in Fp and the polynomial methodmath.univ-lille1.fr/~bhowmik/seminaire... · Additive combinatorics in Fp and the polynomial method The Combinatorial Nullstellensatz](https://reader034.fdocuments.in/reader034/viewer/2022050111/5f48d3be62c3da5adb64be94/html5/thumbnails/29.jpg)
Additive combinatorics in Fp and the polynomial method
Addition Theorems
Set of Subsums
(X0 + · · ·+ Xd−1)t
∏06i<j6d−1
(X 2
j − X 2i
)
=
∑(t0,...,td−1)∑d−1
i=0 ti=t
t!∏d−1i=0 ti !
d−1∏i=0
X tii
∑σ∈Sd
sign(σ)d−1∏i=0
X2σ(i)i
It suffices to prove that the following binomial determinant isnon zero:
Dd ,i0 =
(d − 1, d , . . . , d − 2 + i0, d + i0, . . . , 2d − 1
0, 2, . . . , 2(i0 − 1), 2i0, . . . , 2(d − 1)
).
![Page 30: Additive combinatorics in Fp and the polynomial methodmath.univ-lille1.fr/~bhowmik/seminaire... · Additive combinatorics in Fp and the polynomial method The Combinatorial Nullstellensatz](https://reader034.fdocuments.in/reader034/viewer/2022050111/5f48d3be62c3da5adb64be94/html5/thumbnails/30.jpg)
Additive combinatorics in Fp and the polynomial method
Addition Theorems
Set of Subsums
(X0 + · · ·+ Xd−1)t
∏06i<j6d−1
(X 2
j − X 2i
)= t!
∑σ∈Sd
sign(σ)∑
(t0,...,td−1)∑d−1i=0 ti=t
1∏d−1i=0 ti !
d−1∏i=0
Xti+2σ(i)i
It suffices to prove that the following binomial determinant isnon zero:
Dd ,i0 =
(d − 1, d , . . . , d − 2 + i0, d + i0, . . . , 2d − 1
0, 2, . . . , 2(i0 − 1), 2i0, . . . , 2(d − 1)
).
![Page 31: Additive combinatorics in Fp and the polynomial methodmath.univ-lille1.fr/~bhowmik/seminaire... · Additive combinatorics in Fp and the polynomial method The Combinatorial Nullstellensatz](https://reader034.fdocuments.in/reader034/viewer/2022050111/5f48d3be62c3da5adb64be94/html5/thumbnails/31.jpg)
Additive combinatorics in Fp and the polynomial method
Addition Theorems
Set of Subsums
(X0 + · · ·+ Xd−1)t
∏06i<j6d−1
(X 2
j − X 2i
)= t!
∑σ∈Sd
sign(σ)∑
(b0,...,bd−1)06bi−2σ(i)6t∑d−1i=0 bi=t+d(d−1)
1∏d−1i=0 (bi − 2σ(i))!
d−1∏i=0
X bii
It suffices to prove that the following binomial determinant is nonzero:
Dd ,i0 =
(d − 1, d , . . . , d − 2 + i0, d + i0, . . . , 2d − 1
0, 2, . . . , 2(i0 − 1), 2i0, . . . , 2(d − 1)
).
![Page 32: Additive combinatorics in Fp and the polynomial methodmath.univ-lille1.fr/~bhowmik/seminaire... · Additive combinatorics in Fp and the polynomial method The Combinatorial Nullstellensatz](https://reader034.fdocuments.in/reader034/viewer/2022050111/5f48d3be62c3da5adb64be94/html5/thumbnails/32.jpg)
Additive combinatorics in Fp and the polynomial method
Addition Theorems
Set of Subsums
(X0 + · · ·+ Xd−1)t
∏06i<j6d−1
(X 2
j − X 2i
)=
∑(b0,...,bd−1)∑d−1
i=0 bi=t+d(d−1)max{bi}<p
(t!∏d−1
i=0 (2i)!∏d−1i=0 bi !
∑σ∈Sd
sign(σ)d−1∏i=0
(bi
2σ(i)
)) d−1∏i=0
X bii +Sp
It suffices to prove that the following binomial determinant is nonzero:
Dd ,i0 =
(d − 1, d , . . . , d − 2 + i0, d + i0, . . . , 2d − 1
0, 2, . . . , 2(i0 − 1), 2i0, . . . , 2(d − 1)
).
![Page 33: Additive combinatorics in Fp and the polynomial methodmath.univ-lille1.fr/~bhowmik/seminaire... · Additive combinatorics in Fp and the polynomial method The Combinatorial Nullstellensatz](https://reader034.fdocuments.in/reader034/viewer/2022050111/5f48d3be62c3da5adb64be94/html5/thumbnails/33.jpg)
Additive combinatorics in Fp and the polynomial method
Addition Theorems
Set of Subsums
(X0 + · · ·+ Xd−1)t
∏06i<j6d−1
(X 2
j − X 2i
)=
∑(b0,...,bd−1)∑d−1
i=0 bi=t+d(d−1)max{bi}<p
(t!∏d−1
i=0 (2i)!∏d−1i=0 bi !
(b0, b1, . . . , bd−1
0, 2, . . . , 2(d − 1)
)) d−1∏i=0
X bii +Sp
It suffices to prove that the following binomial determinant is nonzero:
Dd ,i0 =
(d − 1, d , . . . , d − 2 + i0, d + i0, . . . , 2d − 1
0, 2, . . . , 2(i0 − 1), 2i0, . . . , 2(d − 1)
).
![Page 34: Additive combinatorics in Fp and the polynomial methodmath.univ-lille1.fr/~bhowmik/seminaire... · Additive combinatorics in Fp and the polynomial method The Combinatorial Nullstellensatz](https://reader034.fdocuments.in/reader034/viewer/2022050111/5f48d3be62c3da5adb64be94/html5/thumbnails/34.jpg)
Additive combinatorics in Fp and the polynomial method
Addition Theorems
Set of Subsums
(X0 + · · ·+ Xd−1)t
∏06i<j6d−1
(X 2
j − X 2i
)=
∑(b0,...,bd−1)∑d−1
i=0 bi=t+d(d−1)max{bi}<p
(t!∏d−1
i=0 (2i)!∏d−1i=0 bi !
(b0, b1, . . . , bd−1
0, 2, . . . , 2(d − 1)
)) d−1∏i=0
X bii +Sp
It suffices to prove that the following binomial determinant is nonzero:
Dd ,i0 =
(d − 1, d , . . . , d − 2 + i0, d + i0, . . . , 2d − 1
0, 2, . . . , 2(i0 − 1), 2i0, . . . , 2(d − 1)
).
![Page 35: Additive combinatorics in Fp and the polynomial methodmath.univ-lille1.fr/~bhowmik/seminaire... · Additive combinatorics in Fp and the polynomial method The Combinatorial Nullstellensatz](https://reader034.fdocuments.in/reader034/viewer/2022050111/5f48d3be62c3da5adb64be94/html5/thumbnails/35.jpg)
Additive combinatorics in Fp and the polynomial method
Addition Theorems
Binomial Determinants
Binomial Determinants
(a1, . . . , ad
b1, . . . , bd
)=
∣∣∣∣∣∣∣∣∣
(a1b1
) (a1b2
). . .
(a1bd
)(a2b1
) (a2b2
). . .
(a2bd
)...
......(ad
b1
) (adb2
). . .
( adbd )
)∣∣∣∣∣∣∣∣∣ .
Dn,h,i =
(n − h − 1, n − h, . . . , n − h + i − 2, n − h + i , . . . , n − 1
0 , 1 , . . . , (i − 1) , i , . . . , (h − 1)
)
Dd ,i =
(d − 1, d , . . . , d − 2 + i , d + i , . . . , 2d − 1
0, 2, . . . , 2(i − 1), 2i , . . . , 2(d − 1)
).
![Page 36: Additive combinatorics in Fp and the polynomial methodmath.univ-lille1.fr/~bhowmik/seminaire... · Additive combinatorics in Fp and the polynomial method The Combinatorial Nullstellensatz](https://reader034.fdocuments.in/reader034/viewer/2022050111/5f48d3be62c3da5adb64be94/html5/thumbnails/36.jpg)
Additive combinatorics in Fp and the polynomial method
Addition Theorems
Binomial Determinants
Binomial Determinants
(a1, . . . , ad
b1, . . . , bd
)=
∣∣∣∣∣∣∣∣∣
(a1b1
) (a1b2
). . .
(a1bd
)(a2b1
) (a2b2
). . .
(a2bd
)...
......(ad
b1
) (adb2
). . .
( adbd )
)∣∣∣∣∣∣∣∣∣ .
Dn,h,i =
(n − h − 1, n − h, . . . , n − h + i − 2, n − h + i , . . . , n − 1
0 , 1 , . . . , (i − 1) , i , . . . , (h − 1)
)
Dd ,i =
(d − 1, d , . . . , d − 2 + i , d + i , . . . , 2d − 1
0, 2, . . . , 2(i − 1), 2i , . . . , 2(d − 1)
).
![Page 37: Additive combinatorics in Fp and the polynomial methodmath.univ-lille1.fr/~bhowmik/seminaire... · Additive combinatorics in Fp and the polynomial method The Combinatorial Nullstellensatz](https://reader034.fdocuments.in/reader034/viewer/2022050111/5f48d3be62c3da5adb64be94/html5/thumbnails/37.jpg)
Additive combinatorics in Fp and the polynomial method
Addition Theorems
Binomial Determinants
Dn,h,i =
(h
i
),
This proves Dias da Silva-Hamidoune Theorem.
Proposition
Let d > 1, one has:Dd ,0 = 2d(d−1)/2,
Dd ,d = 2(d−1)(d−2)/2.
Dd ,i = 2d−1 d − 1
d − 2 + iDd−1,i−1 + 2d−1 d − 1
d − 1 + iDd−1,i .
Dd ,i = 2d(d−1)
2−i
(d
i
)d + i
d.
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Additive combinatorics in Fp and the polynomial method
Addition Theorems
Binomial Determinants
Dn,h,i =
(h
i
),
This proves Dias da Silva-Hamidoune Theorem.
Proposition
Let d > 1, one has:Dd ,0 = 2d(d−1)/2,
Dd ,d = 2(d−1)(d−2)/2.
Dd ,i = 2d−1 d − 1
d − 2 + iDd−1,i−1 + 2d−1 d − 1
d − 1 + iDd−1,i .
Dd ,i = 2d(d−1)
2−i
(d
i
)d + i
d.
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Additive combinatorics in Fp and the polynomial method
Addition Theorems
Binomial Determinants
Dn,h,i =
(h
i
),
This proves Dias da Silva-Hamidoune Theorem.
Proposition
Let d > 1, one has:Dd ,0 = 2d(d−1)/2,
Dd ,d = 2(d−1)(d−2)/2.
Dd ,i = 2d−1 d − 1
d − 2 + iDd−1,i−1 + 2d−1 d − 1
d − 1 + iDd−1,i .
Dd ,i = 2d(d−1)
2−i
(d
i
)d + i
d.
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Additive combinatorics in Fp and the polynomial method
Addition Theorems
Binomial Determinants
Dn,h,i =
(h
i
),
This proves Dias da Silva-Hamidoune Theorem.
Proposition
Let d > 1, one has:Dd ,0 = 2d(d−1)/2,
Dd ,d = 2(d−1)(d−2)/2.
Dd ,i = 2d−1 d − 1
d − 2 + iDd−1,i−1 + 2d−1 d − 1
d − 1 + iDd−1,i .
Dd ,i = 2d(d−1)
2−i
(d
i
)d + i
d.
![Page 41: Additive combinatorics in Fp and the polynomial methodmath.univ-lille1.fr/~bhowmik/seminaire... · Additive combinatorics in Fp and the polynomial method The Combinatorial Nullstellensatz](https://reader034.fdocuments.in/reader034/viewer/2022050111/5f48d3be62c3da5adb64be94/html5/thumbnails/41.jpg)
Additive combinatorics in Fp and the polynomial method
Additive results on sequences
Three Additive results on sequences in Fp
I Erdos-Ginzburg-Ziv
I Snevily’s conjecture (Arsovsky)
I Nullstellensatz for sequences
![Page 42: Additive combinatorics in Fp and the polynomial methodmath.univ-lille1.fr/~bhowmik/seminaire... · Additive combinatorics in Fp and the polynomial method The Combinatorial Nullstellensatz](https://reader034.fdocuments.in/reader034/viewer/2022050111/5f48d3be62c3da5adb64be94/html5/thumbnails/42.jpg)
Additive combinatorics in Fp and the polynomial method
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. Ifone considers some sets Si , i = 1..n, |Si | = 2. There existss = (s1, . . . , sn) ∈ S1 × · · · × Sn, such that As and b arecoordoninatewise distincts.
proof: The polynomial
n∏i=1
n∑j=1
ai ,jXj − bi
has degree n and the coefficient of
∏ni=1 Xi is Per(A) 6= 0.
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Additive combinatorics in Fp and the polynomial method
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. Ifone considers some sets Si , i = 1..n, |Si | = 2. There existss = (s1, . . . , sn) ∈ S1 × · · · × Sn, such that As and b arecoordoninatewise distincts.
proof: The polynomial
n∏i=1
n∑j=1
ai ,jXj − bi
has degree n and the coefficient of
∏ni=1 Xi is Per(A) 6= 0.
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Additive combinatorics in Fp and the polynomial method
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.
In [0,p-1], g1 6 g2 6 · · · 6 g2p−1.
I If gi = gi+p−1, one has p equal elements.
I Otherwise, one considers the (p − 1)× (p − 1) matrix withonly 1’s, its permanent is (p − 1)! 6= 0. DefineSi = {gi , gi+p−1}, for i ∈ [1, p − 1], of cardinality 2, and bcontaining all values but −g2p−1.
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Additive combinatorics in Fp and the polynomial method
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.
In [0,p-1], g1 6 g2 6 · · · 6 g2p−1.
I If gi = gi+p−1, one has p equal elements.
I Otherwise, one considers the (p − 1)× (p − 1) matrix withonly 1’s, its permanent is (p − 1)! 6= 0. DefineSi = {gi , gi+p−1}, for i ∈ [1, p − 1], of cardinality 2, and bcontaining all values but −g2p−1.
![Page 46: Additive combinatorics in Fp and the polynomial methodmath.univ-lille1.fr/~bhowmik/seminaire... · Additive combinatorics in Fp and the polynomial method The Combinatorial Nullstellensatz](https://reader034.fdocuments.in/reader034/viewer/2022050111/5f48d3be62c3da5adb64be94/html5/thumbnails/46.jpg)
Additive combinatorics in Fp and the polynomial method
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.
In [0,p-1], g1 6 g2 6 · · · 6 g2p−1.
I If gi = gi+p−1, one has p equal elements.
I Otherwise, one considers the (p − 1)× (p − 1) matrix withonly 1’s, its permanent is (p − 1)! 6= 0. DefineSi = {gi , gi+p−1}, for i ∈ [1, p − 1], of cardinality 2, and bcontaining all values but −g2p−1.
![Page 47: Additive combinatorics in Fp and the polynomial methodmath.univ-lille1.fr/~bhowmik/seminaire... · Additive combinatorics in Fp and the polynomial method The Combinatorial Nullstellensatz](https://reader034.fdocuments.in/reader034/viewer/2022050111/5f48d3be62c3da5adb64be94/html5/thumbnails/47.jpg)
Additive combinatorics in Fp and the polynomial method
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.
In [0,p-1], g1 6 g2 6 · · · 6 g2p−1.
I If gi = gi+p−1, one has p equal elements.
I Otherwise, one considers the (p − 1)× (p − 1) matrix withonly 1’s, its permanent is (p − 1)! 6= 0. DefineSi = {gi , gi+p−1}, for i ∈ [1, p − 1], of cardinality 2, and bcontaining all values but −g2p−1.
![Page 48: Additive combinatorics in Fp and the polynomial methodmath.univ-lille1.fr/~bhowmik/seminaire... · Additive combinatorics in Fp and the polynomial method The Combinatorial Nullstellensatz](https://reader034.fdocuments.in/reader034/viewer/2022050111/5f48d3be62c3da5adb64be94/html5/thumbnails/48.jpg)
Additive combinatorics in Fp and the polynomial method
Additive results on sequences
Snevily’s conjecture
Snevily’s ConjectureG a finite group of odd order.a1, . . . , ak , k distinct elements and b1, . . . , bk , k distincts elements.then there exists a permutation π of [1, k] such thata1 + bπ(1), . . . , ak + bπ(k) are pairwise distincts.
I G = Z/pZ (Alon - 2000)
I G = Z/nZ (Dasgupta, Karolyi, Serra, Szegedy - 2001),in F2d (with n | 2d − 1), g of order n in F×
2d , the polynomial:
P(X1, . . . ,Xk) =∏
16j<i6k
(Xi − Xj) (αiXi − αjXj) .
has degree k(k − 1), where αi = gbi andA1 = · · · = Ak = {gai | i = 1..k}.
Coefficient of∏k
i=1 X k−1i is the Van der Monde permanent of
the αi ’s. In F2d , permanent and determinant are equal.
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Additive combinatorics in Fp and the polynomial method
Additive results on sequences
Snevily’s conjecture
Snevily’s ConjectureG a finite group of odd order.a1, . . . , ak , k distinct elements and b1, . . . , bk , k distincts elements.then there exists a permutation π of [1, k] such thata1 + bπ(1), . . . , ak + bπ(k) are pairwise distincts.
I G = Z/pZ (Alon - 2000)
I G = Z/nZ (Dasgupta, Karolyi, Serra, Szegedy - 2001),in F2d (with n | 2d − 1), g of order n in F×
2d , the polynomial:
P(X1, . . . ,Xk) =∏
16j<i6k
(Xi − Xj) (αiXi − αjXj) .
has degree k(k − 1), where αi = gbi andA1 = · · · = Ak = {gai | i = 1..k}.
Coefficient of∏k
i=1 X k−1i is the Van der Monde permanent of
the αi ’s. In F2d , permanent and determinant are equal.
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Additive combinatorics in Fp and the polynomial method
Additive results on sequences
Snevily’s conjecture
Snevily’s ConjectureG a finite group of odd order.a1, . . . , ak , k distinct elements and b1, . . . , bk , k distincts elements.then there exists a permutation π of [1, k] such thata1 + bπ(1), . . . , ak + bπ(k) are pairwise distincts.
I G = Z/pZ (Alon - 2000)
I G = Z/nZ (Dasgupta, Karolyi, Serra, Szegedy - 2001),in F2d (with n | 2d − 1), g of order n in F×
2d , the polynomial:
P(X1, . . . ,Xk) =∏
16j<i6k
(Xi − Xj) (αiXi − αjXj) .
has degree k(k − 1), where αi = gbi andA1 = · · · = Ak = {gai | i = 1..k}.
Coefficient of∏k
i=1 X k−1i is the Van der Monde permanent of
the αi ’s. In F2d , permanent and determinant are equal.
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Additive combinatorics in Fp and the polynomial method
Additive results on sequences
Snevily’s conjecture
Snevily’s ConjectureG a finite group of odd order.a1, . . . , ak , k distinct elements and b1, . . . , bk , k distincts elements.then there exists a permutation π of [1, k] such thata1 + bπ(1), . . . , ak + bπ(k) are pairwise distincts.
I G = Z/pZ (Alon - 2000)
I G = Z/nZ (Dasgupta, Karolyi, Serra, Szegedy - 2001),in F2d (with n | 2d − 1), g of order n in F×
2d , the polynomial:
P(X1, . . . ,Xk) =∏
16j<i6k
(Xi − Xj) (αiXi − αjXj) .
has degree k(k − 1), where αi = gbi andA1 = · · · = Ak = {gai | i = 1..k}.Coefficient of
∏ki=1 X k−1
i is the Van der Monde permanent ofthe αi ’s. In F2d , permanent and determinant are equal.
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Additive combinatorics in Fp and the polynomial method
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}`
One considers:dim(A) = dim(〈SA〉).
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Additive combinatorics in Fp and the polynomial method
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}`
One considers:dim(A) = dim(〈SA〉).
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Additive combinatorics in Fp and the polynomial method
Additive results on sequences
Nullstellensatz for sequences
Theorem (B.-Girard)
p a prime number, A = (a1, . . . , a`) a sequence of ` > p elementsof F×p :
dim(A) = `− 1.
Method: SA ⊂ SB , Iλ = {i ∈ [1, `] : bi/ai = λ} .λ1, . . . , λd ratios associated to (A,B) and Si = (aj : j ∈ Iλi
) andΣi = Σ(Si ). The polynomial
P(X1, . . . ,Xd) =
(d∑
i=1
λiXi
)( d∑i=1
Xi
)p−1
− 1
,
vanishes on∏d
i=1 Σi , coefficient of∏
X tii is c
∑λi ti .
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Additive combinatorics in Fp and the polynomial method
Additive results on sequences
Nullstellensatz for sequences
Theorem (B.-Girard)
p a prime number, A = (a1, . . . , a`) a sequence of ` > p elementsof F×p :
dim(A) = `− 1.
Method: SA ⊂ SB , Iλ = {i ∈ [1, `] : bi/ai = λ} .λ1, . . . , λd ratios associated to (A,B) and Si = (aj : j ∈ Iλi
) andΣi = Σ(Si ). The polynomial
P(X1, . . . ,Xd) =
(d∑
i=1
λiXi
)( d∑i=1
Xi
)p−1
− 1
,
vanishes on∏d
i=1 Σi , coefficient of∏
X tii is c
∑λi ti .
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Additive combinatorics in Fp and the polynomial method
Additive results on sequences
Nullstellensatz for sequences
Theorem (B.-Girard)
p a prime number, A = (a1, . . . , a`) a sequence of ` > p elementsof F×p :
dim(A) = `− 1.
Method: SA ⊂ SB , Iλ = {i ∈ [1, `] : bi/ai = λ} .
λ1, . . . , λd ratios associated to (A,B) and Si = (aj : j ∈ Iλi) and
Σi = Σ(Si ). The polynomial
P(X1, . . . ,Xd) =
(d∑
i=1
λiXi
)( d∑i=1
Xi
)p−1
− 1
,
vanishes on∏d
i=1 Σi , coefficient of∏
X tii is c
∑λi ti .
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Additive combinatorics in Fp and the polynomial method
Additive results on sequences
Nullstellensatz for sequences
Theorem (B.-Girard)
p a prime number, A = (a1, . . . , a`) a sequence of ` > p elementsof F×p :
dim(A) = `− 1.
Method: SA ⊂ SB , Iλ = {i ∈ [1, `] : bi/ai = λ} .λ1, . . . , λd ratios associated to (A,B) and Si = (aj : j ∈ Iλi
) andΣi = Σ(Si ).
The polynomial
P(X1, . . . ,Xd) =
(d∑
i=1
λiXi
)( d∑i=1
Xi
)p−1
− 1
,
vanishes on∏d
i=1 Σi , coefficient of∏
X tii is c
∑λi ti .
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Additive combinatorics in Fp and the polynomial method
Additive results on sequences
Nullstellensatz for sequences
Theorem (B.-Girard)
p a prime number, A = (a1, . . . , a`) a sequence of ` > p elementsof F×p :
dim(A) = `− 1.
Method: SA ⊂ SB , Iλ = {i ∈ [1, `] : bi/ai = λ} .λ1, . . . , λd ratios associated to (A,B) and Si = (aj : j ∈ Iλi
) andΣi = Σ(Si ). The polynomial
P(X1, . . . ,Xd) =
(d∑
i=1
λiXi
)( d∑i=1
Xi
)p−1
− 1
,
vanishes on∏d
i=1 Σi , coefficient of∏
X tii is c
∑λi ti .
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Additive combinatorics in Fp and the polynomial method
Additive results on sequences
Nullstellensatz for sequences
Theorem (B.-Girard)
p a prime number, A = (a1, . . . , ap) a sequence of p elements ofF×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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Additive combinatorics in Fp and the polynomial method
Additive results on sequences
Nullstellensatz for sequences
Theorem (B.-Girard)
p a prime number, A = (a1, . . . , ap) a sequence of p elements ofF×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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Additive combinatorics in Fp and the polynomial method
Additive results on sequences
Nullstellensatz for sequences
Theorem (B.-Girard)
p a prime number, A = (a1, . . . , ap) a sequence of p elements ofF×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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Additive combinatorics in Fp and the polynomial method
Additive results on sequences
Nullstellensatz for sequences
Theorem (B.-Girard)
p a prime number, A = (a1, . . . , ap) a sequence of p elements ofF×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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Additive combinatorics in Fp and the polynomial method
Additive results on sequences
Nullstellensatz for sequences
The polynomial
P(X1, . . . ,Xd) =
(d∑
i=1
λiXi
)( d∑i=1
Xi
)p−1
− 1
,
vanishes on∏d
i=1 Σi , the coefficient of∏
X tii is c
∑λi ti .
Id∑
i=1
(|Σi | − 1) ≥ p
I If∑
(|Σi | − 1) = p, one has:∑λi (|Σi | − 1) = 0
I And |Σi | − 1 = |Si |, Σi arithmetic progression.
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Additive combinatorics in Fp and the polynomial method
Additive results on sequences
Nullstellensatz for sequences
The polynomial
P(X1, . . . ,Xd) =
(d∑
i=1
λiXi
)( d∑i=1
Xi
)p−1
− 1
,
vanishes on∏d
i=1 Σi , the coefficient of∏
X tii is c
∑λi ti .
Id∑
i=1
(|Σi | − 1) ≥ p
I If∑
(|Σi | − 1) = p, one has:∑λi (|Σi | − 1) = 0
I And |Σi | − 1 = |Si |, Σi arithmetic progression.
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Additive combinatorics in Fp and the polynomial method
Additive results on sequences
Nullstellensatz for sequences
The polynomial
P(X1, . . . ,Xd) =
(d∑
i=1
λiXi
)( d∑i=1
Xi
)p−1
− 1
,
vanishes on∏d
i=1 Σi , the coefficient of∏
X tii is c
∑λi ti .
Id∑
i=1
(|Σi | − 1) ≥ p
I If∑
(|Σi | − 1) = p, one has:∑λi (|Σi | − 1) = 0
I And |Σi | − 1 = |Si |, Σi arithmetic progression.
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Additive combinatorics in Fp and the polynomial method
Additive results on sequences
Nullstellensatz for sequences
The polynomial
P(X1, . . . ,Xd) =
(d∑
i=1
λiXi
)( d∑i=1
Xi
)p−1
− 1
,
vanishes on∏d
i=1 Σi , the coefficient of∏
X tii is c
∑λi ti .
Id∑
i=1
(|Σi | − 1) ≥ p
I If∑
(|Σi | − 1) = p, one has:∑λi (|Σi | − 1) = 0
I And |Σi | − 1 = |Si |, Σi arithmetic progression.
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Additive combinatorics in Fp and the polynomial method
Additive results on sequences
Nullstellensatz for sequences
i0, j0 ∈ [1, d ] and ri0 ∈ Si0
Σ′i0 = Σ(Si0 r (ri0)), Σ′j0 = Σ(Sj0 ∪ (ri0)) = Σj0 + {0, ri0}
Qi0,j0(X1, . . . ,Xd) =
(d∑
i=1
λiXi
)(d∑
i=1
λiXi − χ
)( d∑i=1
Xi
)p−1
− 1
,
vanishes on∏d
i=1 Σ′i , χ = (λj0 − λi0)ri0 . |Σ′i0 | = |Σi0 | − 1 et|Σ′j0 | ≥ |Σj0 |+ 2For ti0 = |Σi0 | − 2, tj0 = |Σj0 |+ 1, one has ti ≤ |Σ′i | − 1, thecoefficient is
2(λj0 − λi0)2 −d∑
i=1
λ2i (|Σi | − 1)
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Additive combinatorics in Fp and the polynomial method
Additive results on sequences
Nullstellensatz for sequences
i0, j0 ∈ [1, d ] and ri0 ∈ Si0
Σ′i0 = Σ(Si0 r (ri0)), Σ′j0 = Σ(Sj0 ∪ (ri0)) = Σj0 + {0, ri0}
Qi0,j0(X1, . . . ,Xd) =
(d∑
i=1
λiXi
)(d∑
i=1
λiXi − χ
)( d∑i=1
Xi
)p−1
− 1
,
vanishes on∏d
i=1 Σ′i , χ = (λj0 − λi0)ri0 .
|Σ′i0 | = |Σi0 | − 1 et|Σ′j0 | ≥ |Σj0 |+ 2For ti0 = |Σi0 | − 2, tj0 = |Σj0 |+ 1, one has ti ≤ |Σ′i | − 1, thecoefficient is
2(λj0 − λi0)2 −d∑
i=1
λ2i (|Σi | − 1)
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Additive combinatorics in Fp and the polynomial method
Additive results on sequences
Nullstellensatz for sequences
i0, j0 ∈ [1, d ] and ri0 ∈ Si0
Σ′i0 = Σ(Si0 r (ri0)), Σ′j0 = Σ(Sj0 ∪ (ri0)) = Σj0 + {0, ri0}
Qi0,j0(X1, . . . ,Xd) =
(d∑
i=1
λiXi
)(d∑
i=1
λiXi − χ
)( d∑i=1
Xi
)p−1
− 1
,
vanishes on∏d
i=1 Σ′i , χ = (λj0 − λi0)ri0 . |Σ′i0 | = |Σi0 | − 1 et|Σ′j0 | ≥ |Σj0 |+ 2
For ti0 = |Σi0 | − 2, tj0 = |Σj0 |+ 1, one has ti ≤ |Σ′i | − 1, thecoefficient is
2(λj0 − λi0)2 −d∑
i=1
λ2i (|Σi | − 1)
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Additive combinatorics in Fp and the polynomial method
Additive results on sequences
Nullstellensatz for sequences
i0, j0 ∈ [1, d ] and ri0 ∈ Si0
Σ′i0 = Σ(Si0 r (ri0)), Σ′j0 = Σ(Sj0 ∪ (ri0)) = Σj0 + {0, ri0}
Qi0,j0(X1, . . . ,Xd) =
(d∑
i=1
λiXi
)(d∑
i=1
λiXi − χ
)( d∑i=1
Xi
)p−1
− 1
,
vanishes on∏d
i=1 Σ′i , χ = (λj0 − λi0)ri0 . |Σ′i0 | = |Σi0 | − 1 et|Σ′j0 | ≥ |Σj0 |+ 2For ti0 = |Σi0 | − 2, tj0 = |Σj0 |+ 1, one has ti ≤ |Σ′i | − 1, thecoefficient is
2(λj0 − λi0)2 −d∑
i=1
λ2i (|Σi | − 1)
![Page 71: Additive combinatorics in Fp and the polynomial methodmath.univ-lille1.fr/~bhowmik/seminaire... · Additive combinatorics in Fp and the polynomial method The Combinatorial Nullstellensatz](https://reader034.fdocuments.in/reader034/viewer/2022050111/5f48d3be62c3da5adb64be94/html5/thumbnails/71.jpg)
Additive combinatorics in Fp and the polynomial method
Additive results on sequences
Nullstellensatz for sequences
i0, j0 ∈ [1, d ] and ri0 ∈ Si0
Σ′i0 = Σ(Si0 r (ri0)), Σ′j0 = Σ(Sj0 ∪ (ri0)) = Σj0 + {0, ri0}
Qi0,j0(X1, . . . ,Xd) =
(d∑
i=1
λiXi
)(d∑
i=1
λiXi − χ
)( d∑i=1
Xi
)p−1
− 1
,
vanishes on∏d
i=1 Σ′i , χ = (λj0 − λi0)ri0 . |Σ′i0 | = |Σi0 | − 1 et|Σ′j0 | ≥ |Σj0 |+ 2For ti0 = |Σi0 | − 2, tj0 = |Σj0 |+ 1, one has ti ≤ |Σ′i | − 1, thecoefficient is
2(λj0 − λi0)2 −d∑
i=1
λ2i (|Σi | − 1)
![Page 72: Additive combinatorics in Fp and the polynomial methodmath.univ-lille1.fr/~bhowmik/seminaire... · Additive combinatorics in Fp and the polynomial method The Combinatorial Nullstellensatz](https://reader034.fdocuments.in/reader034/viewer/2022050111/5f48d3be62c3da5adb64be94/html5/thumbnails/72.jpg)
Additive combinatorics in Fp and the polynomial method
Additive results on sequences
Nullstellensatz for sequences
Theorem (B.-Girard)
p a prime number, A = (a1, . . . , ap−1) a zerosum sequence ofp − 1 elements of F×p :
I dim(A) = 1,
(r , . . . , r , 2r).
(aσ(1), . . . , aσ(p−1)) = (r , . . . , r , 2r).
I dim(A) = p − 4,
(r , . . . , r︸ ︷︷ ︸p−5
,−r , 2r , 2r , 2r).
I dim(A) = p − 3, ∃t ∈ [0, p − 6],
(r , . . . , r︸ ︷︷ ︸t
,−r , . . . ,−r︸ ︷︷ ︸p−4−t
, 2r ,−(t + 3)r ,−(t + 3)r).
I dim(A) = p − 3, ∃t ∈ [1, p − 4],
(r , . . . , r︸ ︷︷ ︸t
,−r , . . . ,−r︸ ︷︷ ︸p−3−t
,−(t + 1)r ,−(t + 2)r).
I p = 7, dim(A) = p − 3 = 4,
(−1, 1,−2, 2,−3, 3).
I dim(A) = p − 2.
![Page 73: Additive combinatorics in Fp and the polynomial methodmath.univ-lille1.fr/~bhowmik/seminaire... · Additive combinatorics in Fp and the polynomial method The Combinatorial Nullstellensatz](https://reader034.fdocuments.in/reader034/viewer/2022050111/5f48d3be62c3da5adb64be94/html5/thumbnails/73.jpg)
Additive combinatorics in Fp and the polynomial method
Additive results on sequences
Nullstellensatz for sequences
Theorem (B.-Girard)
p a prime number, A = (a1, . . . , ap−1) a zerosum sequence ofp − 1 elements of F×p :
I dim(A) = 1, (r , . . . , r , 2r).
I dim(A) = p − 4,
(r , . . . , r︸ ︷︷ ︸p−5
,−r , 2r , 2r , 2r).
(aσ(1), . . . , aσ(p−1)) = (r , . . . , r︸ ︷︷ ︸p−5
,−r , 2r , 2r , 2r).
I dim(A) = p − 3, ∃t ∈ [0, p − 6],
(r , . . . , r︸ ︷︷ ︸t
,−r , . . . ,−r︸ ︷︷ ︸p−4−t
, 2r ,−(t + 3)r ,−(t + 3)r).
I dim(A) = p − 3, ∃t ∈ [1, p − 4],
(r , . . . , r︸ ︷︷ ︸t
,−r , . . . ,−r︸ ︷︷ ︸p−3−t
,−(t + 1)r ,−(t + 2)r).
I p = 7, dim(A) = p − 3 = 4,
(−1, 1,−2, 2,−3, 3).
I dim(A) = p − 2.
![Page 74: Additive combinatorics in Fp and the polynomial methodmath.univ-lille1.fr/~bhowmik/seminaire... · Additive combinatorics in Fp and the polynomial method The Combinatorial Nullstellensatz](https://reader034.fdocuments.in/reader034/viewer/2022050111/5f48d3be62c3da5adb64be94/html5/thumbnails/74.jpg)
Additive combinatorics in Fp and the polynomial method
Additive results on sequences
Nullstellensatz for sequences
Theorem (B.-Girard)
p a prime number, A = (a1, . . . , ap−1) a zerosum sequence ofp − 1 elements of F×p :
I dim(A) = 1, (r , . . . , r , 2r).
I dim(A) = p − 4, (r , . . . , r︸ ︷︷ ︸p−5
,−r , 2r , 2r , 2r).
I dim(A) = p − 3, ∃t ∈ [0, p − 6],
(r , . . . , r︸ ︷︷ ︸t
,−r , . . . ,−r︸ ︷︷ ︸p−4−t
, 2r ,−(t + 3)r ,−(t + 3)r).
(aσ(1), . . . , aσ(p−1)) = (r , . . . , r︸ ︷︷ ︸t
,−r , . . . ,−r︸ ︷︷ ︸p−4−t
, 2r ,−(t+3)r ,−(t+3)r).
I dim(A) = p − 3, ∃t ∈ [1, p − 4],
(r , . . . , r︸ ︷︷ ︸t
,−r , . . . ,−r︸ ︷︷ ︸p−3−t
,−(t + 1)r ,−(t + 2)r).
I p = 7, dim(A) = p − 3 = 4,
(−1, 1,−2, 2,−3, 3).
I dim(A) = p − 2.
![Page 75: Additive combinatorics in Fp and the polynomial methodmath.univ-lille1.fr/~bhowmik/seminaire... · Additive combinatorics in Fp and the polynomial method The Combinatorial Nullstellensatz](https://reader034.fdocuments.in/reader034/viewer/2022050111/5f48d3be62c3da5adb64be94/html5/thumbnails/75.jpg)
Additive combinatorics in Fp and the polynomial method
Additive results on sequences
Nullstellensatz for sequences
Theorem (B.-Girard)
p a prime number, A = (a1, . . . , ap−1) a zerosum sequence ofp − 1 elements of F×p :
I dim(A) = 1, (r , . . . , r , 2r).
I dim(A) = p − 4, (r , . . . , r︸ ︷︷ ︸p−5
,−r , 2r , 2r , 2r).
I dim(A) = p − 3, ∃t ∈ [0, p − 6],(r , . . . , r︸ ︷︷ ︸
t
,−r , . . . ,−r︸ ︷︷ ︸p−4−t
, 2r ,−(t + 3)r ,−(t + 3)r).
I dim(A) = p − 3, ∃t ∈ [1, p − 4],
(r , . . . , r︸ ︷︷ ︸t
,−r , . . . ,−r︸ ︷︷ ︸p−3−t
,−(t + 1)r ,−(t + 2)r).
(aσ(1), . . . , aσ(p−1)) = (r , . . . , r︸ ︷︷ ︸t
,−r , . . . ,−r︸ ︷︷ ︸p−3−t
,−(t+1)r ,−(t+2)r).
I p = 7, dim(A) = p − 3 = 4,
(−1, 1,−2, 2,−3, 3).
I dim(A) = p − 2.
![Page 76: Additive combinatorics in Fp and the polynomial methodmath.univ-lille1.fr/~bhowmik/seminaire... · Additive combinatorics in Fp and the polynomial method The Combinatorial Nullstellensatz](https://reader034.fdocuments.in/reader034/viewer/2022050111/5f48d3be62c3da5adb64be94/html5/thumbnails/76.jpg)
Additive combinatorics in Fp and the polynomial method
Additive results on sequences
Nullstellensatz for sequences
Theorem (B.-Girard)
p a prime number, A = (a1, . . . , ap−1) a zerosum sequence ofp − 1 elements of F×p :
I dim(A) = 1, (r , . . . , r , 2r).
I dim(A) = p − 4, (r , . . . , r︸ ︷︷ ︸p−5
,−r , 2r , 2r , 2r).
I dim(A) = p − 3, ∃t ∈ [0, p − 6],(r , . . . , r︸ ︷︷ ︸
t
,−r , . . . ,−r︸ ︷︷ ︸p−4−t
, 2r ,−(t + 3)r ,−(t + 3)r).
I dim(A) = p − 3, ∃t ∈ [1, p − 4],(r , . . . , r︸ ︷︷ ︸
t
,−r , . . . ,−r︸ ︷︷ ︸p−3−t
,−(t + 1)r ,−(t + 2)r).
I p = 7, dim(A) = p − 3 = 4,
(−1, 1,−2, 2,−3, 3).
(aσ(1), . . . , aσ(6)) = (−1, 1,−2, 2,−3, 3).
I dim(A) = p − 2.
![Page 77: Additive combinatorics in Fp and the polynomial methodmath.univ-lille1.fr/~bhowmik/seminaire... · Additive combinatorics in Fp and the polynomial method The Combinatorial Nullstellensatz](https://reader034.fdocuments.in/reader034/viewer/2022050111/5f48d3be62c3da5adb64be94/html5/thumbnails/77.jpg)
Additive combinatorics in Fp and the polynomial method
Additive results on sequences
Nullstellensatz for sequences
Theorem (B.-Girard)
p a prime number, A = (a1, . . . , ap−1) a zerosum sequence ofp − 1 elements of F×p :
I dim(A) = 1, (r , . . . , r , 2r).
I dim(A) = p − 4, (r , . . . , r︸ ︷︷ ︸p−5
,−r , 2r , 2r , 2r).
I dim(A) = p − 3, ∃t ∈ [0, p − 6],(r , . . . , r︸ ︷︷ ︸
t
,−r , . . . ,−r︸ ︷︷ ︸p−4−t
, 2r ,−(t + 3)r ,−(t + 3)r).
I dim(A) = p − 3, ∃t ∈ [1, p − 4],(r , . . . , r︸ ︷︷ ︸
t
,−r , . . . ,−r︸ ︷︷ ︸p−3−t
,−(t + 1)r ,−(t + 2)r).
I p = 7, dim(A) = p − 3 = 4, (−1, 1,−2, 2,−3, 3).
I dim(A) = p − 2.