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MUIC Applied Math Seminar
Number Theory in Function Fields
Chatchawan Panraksa
Science DivisionMahidol University International College
October 9, 2019
FToA
TheoremLet f (x) ∈ C[x ]. Then f (x) has at least one complex root.
C. Panraksa (MUIC) Number Theory in Function Fields October 9, 2019 2 / 14
Theorem (Combinatorial Nullstellensatz, Noga Alon, 1999)Let F be a field, and let f = f (x1, . . . , xn) be a polynomial in F [x1, . . . , xn].
Suppose the degree deg(f ) of f isn∑
i=1
ti , where each ti is a nonnegative
integer, and suppose the coefficient ofn∏
i=1
x tii in f is nonzero. Then, if
S1, . . . ,Sn are subsets of F with |Si | > ti , there ares1 ∈ S1, s2 ∈ S2, . . . , sn ∈ Sn so that
f (x1, . . . , sn) 6= 0.
C. Panraksa (MUIC) Number Theory in Function Fields October 9, 2019 3 / 14
Cauchy-Davenport Theorem
Theorem (Cauchy-Davenport)Given A,B non-empty subsets of Zp for a prime p, then
|A+ B| ≥ min{p, |A|+ |B| − 1}.
C. Panraksa (MUIC) Number Theory in Function Fields October 9, 2019 4 / 14
(Almost) the Hardest IMO!
C. Panraksa (MUIC) Number Theory in Function Fields October 9, 2019 5 / 14
(Almost) the Hardest IMO!
C. Panraksa (MUIC) Number Theory in Function Fields October 9, 2019 6 / 14
IMO 2007 Problem 6
IMO 2007 Problem 6: Let n be a positive integer. Consider
S = {(x , y , z) | x , y , z ∈ {0, 1, . . . , n}, x + y + z > 0}
as a set of (n+ 1)3− 1 points in 3D space. Determine the smallest numberof planes, the union of which contains S but does not include (0, 0, 0).
C. Panraksa (MUIC) Number Theory in Function Fields October 9, 2019 7 / 14
3D Lattices
(0,0,0)
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DefinitionA radical of a positive integer is a product of its distinct prime factors.NOTATION: n = pk1
1 pk11 . . . pkmm
radical of n is rad(n) = p1p2 . . . pm.rad(1) := 1
Examplerad(5) = 5rad(345744) = rad(24 · 32 · 74) = 2 · 3 · 7 = 42rad(1868347265625) = rad(314 · 58) = 3 · 5 = 15
C. Panraksa (MUIC) Number Theory in Function Fields October 9, 2019 9 / 14
DefinitionA radical of a positive integer is a product of its distinct prime factors.NOTATION: n = pk1
1 pk11 . . . pkmm
radical of n is rad(n) = p1p2 . . . pm.rad(1) := 1
Examplerad(5) = 5
rad(345744) = rad(24 · 32 · 74) = 2 · 3 · 7 = 42rad(1868347265625) = rad(314 · 58) = 3 · 5 = 15
C. Panraksa (MUIC) Number Theory in Function Fields October 9, 2019 9 / 14
DefinitionA radical of a positive integer is a product of its distinct prime factors.NOTATION: n = pk1
1 pk11 . . . pkmm
radical of n is rad(n) = p1p2 . . . pm.rad(1) := 1
Examplerad(5) = 5rad(345744)
= rad(24 · 32 · 74) = 2 · 3 · 7 = 42rad(1868347265625) = rad(314 · 58) = 3 · 5 = 15
C. Panraksa (MUIC) Number Theory in Function Fields October 9, 2019 9 / 14
DefinitionA radical of a positive integer is a product of its distinct prime factors.NOTATION: n = pk1
1 pk11 . . . pkmm
radical of n is rad(n) = p1p2 . . . pm.rad(1) := 1
Examplerad(5) = 5rad(345744) = rad(24 · 32 · 74)
= 2 · 3 · 7 = 42rad(1868347265625) = rad(314 · 58) = 3 · 5 = 15
C. Panraksa (MUIC) Number Theory in Function Fields October 9, 2019 9 / 14
DefinitionA radical of a positive integer is a product of its distinct prime factors.NOTATION: n = pk1
1 pk11 . . . pkmm
radical of n is rad(n) = p1p2 . . . pm.rad(1) := 1
Examplerad(5) = 5rad(345744) = rad(24 · 32 · 74) = 2 · 3 · 7
= 42rad(1868347265625) = rad(314 · 58) = 3 · 5 = 15
C. Panraksa (MUIC) Number Theory in Function Fields October 9, 2019 9 / 14
DefinitionA radical of a positive integer is a product of its distinct prime factors.NOTATION: n = pk1
1 pk11 . . . pkmm
radical of n is rad(n) = p1p2 . . . pm.rad(1) := 1
Examplerad(5) = 5rad(345744) = rad(24 · 32 · 74) = 2 · 3 · 7 = 42
rad(1868347265625) = rad(314 · 58) = 3 · 5 = 15
C. Panraksa (MUIC) Number Theory in Function Fields October 9, 2019 9 / 14
DefinitionA radical of a positive integer is a product of its distinct prime factors.NOTATION: n = pk1
1 pk11 . . . pkmm
radical of n is rad(n) = p1p2 . . . pm.rad(1) := 1
Examplerad(5) = 5rad(345744) = rad(24 · 32 · 74) = 2 · 3 · 7 = 42rad(1868347265625)
= rad(314 · 58) = 3 · 5 = 15
C. Panraksa (MUIC) Number Theory in Function Fields October 9, 2019 9 / 14
DefinitionA radical of a positive integer is a product of its distinct prime factors.NOTATION: n = pk1
1 pk11 . . . pkmm
radical of n is rad(n) = p1p2 . . . pm.rad(1) := 1
Examplerad(5) = 5rad(345744) = rad(24 · 32 · 74) = 2 · 3 · 7 = 42rad(1868347265625) = rad(314 · 58)
= 3 · 5 = 15
C. Panraksa (MUIC) Number Theory in Function Fields October 9, 2019 9 / 14
DefinitionA radical of a positive integer is a product of its distinct prime factors.NOTATION: n = pk1
1 pk11 . . . pkmm
radical of n is rad(n) = p1p2 . . . pm.rad(1) := 1
Examplerad(5) = 5rad(345744) = rad(24 · 32 · 74) = 2 · 3 · 7 = 42rad(1868347265625) = rad(314 · 58) = 3 · 5
= 15
C. Panraksa (MUIC) Number Theory in Function Fields October 9, 2019 9 / 14
DefinitionA radical of a positive integer is a product of its distinct prime factors.NOTATION: n = pk1
1 pk11 . . . pkmm
radical of n is rad(n) = p1p2 . . . pm.rad(1) := 1
Examplerad(5) = 5rad(345744) = rad(24 · 32 · 74) = 2 · 3 · 7 = 42rad(1868347265625) = rad(314 · 58) = 3 · 5 = 15
C. Panraksa (MUIC) Number Theory in Function Fields October 9, 2019 9 / 14
How about raising the power of rad(abc)?
Conjecture (abc−Conjecture, Masser and Oestelé 1988))For every ε > 0, there are only finite many triple (a, b, c) of coprimepositve integers with a+ b = c such that:
c > rad(abc)1+ε.
Equivalently, we have
Conjecture (abc−Conjecture)For every ε > 0, there exists a constant Kε such that for all triples (a, b, c)of coprime positve integers, with a+ b = c such that
c < Kεrad(abc)1+ε.
C. Panraksa (MUIC) Number Theory in Function Fields October 9, 2019 10 / 14
How about raising the power of rad(abc)?
Conjecture (abc−Conjecture, Masser and Oestelé 1988))For every ε > 0, there are only finite many triple (a, b, c) of coprimepositve integers with a+ b = c such that:
c > rad(abc)1+ε.
Equivalently, we have
Conjecture (abc−Conjecture)For every ε > 0, there exists a constant Kε such that for all triples (a, b, c)of coprime positve integers, with a+ b = c such that
c < Kεrad(abc)1+ε.
C. Panraksa (MUIC) Number Theory in Function Fields October 9, 2019 10 / 14
abc Theorem for Function Fields
Theorem (Mason-Stothers Theorem, 1981)Let K be a field of characteristic 0. If a(t), b(t), c(t) are nonzeropolynomials in K [t] with a(t) + b(t) = c(t) and gcd(a(t), b(t), c(t)) = 1,then
max{deg a, deg b, deg c} ≤ rad(abc)− 1.
Example
a(t) = 1, b(t) = tn, c(t) = tn + 1
max{deg a, deg b, deg c} = nrad(abc) = t(tn + 1), deg rad(abc) = n + 1
C. Panraksa (MUIC) Number Theory in Function Fields October 9, 2019 11 / 14
abc Theorem for Function Fields
Theorem (Mason-Stothers Theorem, 1981)Let K be a field of characteristic 0. If a(t), b(t), c(t) are nonzeropolynomials in K [t] with a(t) + b(t) = c(t) and gcd(a(t), b(t), c(t)) = 1,then
max{deg a, deg b, deg c} ≤ rad(abc)− 1.
Example
a(t) = 1, b(t) = tn, c(t) = tn + 1max{deg a, deg b, deg c} = n
rad(abc) = t(tn + 1), deg rad(abc) = n + 1
C. Panraksa (MUIC) Number Theory in Function Fields October 9, 2019 11 / 14
abc Theorem for Function Fields
Theorem (Mason-Stothers Theorem, 1981)Let K be a field of characteristic 0. If a(t), b(t), c(t) are nonzeropolynomials in K [t] with a(t) + b(t) = c(t) and gcd(a(t), b(t), c(t)) = 1,then
max{deg a, deg b, deg c} ≤ rad(abc)− 1.
Example
a(t) = 1, b(t) = tn, c(t) = tn + 1max{deg a, deg b, deg c} = nrad(abc) = t(tn + 1),
deg rad(abc) = n + 1
C. Panraksa (MUIC) Number Theory in Function Fields October 9, 2019 11 / 14
abc Theorem for Function Fields
Theorem (Mason-Stothers Theorem, 1981)Let K be a field of characteristic 0. If a(t), b(t), c(t) are nonzeropolynomials in K [t] with a(t) + b(t) = c(t) and gcd(a(t), b(t), c(t)) = 1,then
max{deg a, deg b, deg c} ≤ rad(abc)− 1.
Example
a(t) = 1, b(t) = tn, c(t) = tn + 1max{deg a, deg b, deg c} = nrad(abc) = t(tn + 1), deg rad(abc) = n + 1
C. Panraksa (MUIC) Number Theory in Function Fields October 9, 2019 11 / 14
Fermat’s Last Theorem for Polynomials
TheoremThere are no nonzero polynomials x(t), y(t), z(t) ∈ C[t] such that
x(t)n + y(t)n = z(t)n, for n > 2.
C. Panraksa (MUIC) Number Theory in Function Fields October 9, 2019 12 / 14
“Mathematicians are like Frenchmen: whatever you say tothem they translate into their own language and forthwith it
is something entirely different.”
Johann Wolfgang von Goethe, 1749 – 1832
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Thank You!
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