elementary problems in number theory - univerzita karlovaical/presentations/szabo.pdf ·...
TRANSCRIPT
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Elementary problems in number theory
Csaba Szabo
Eotvos Lorand University, Budapest
June, 2010
Csaba Szabo Elementary problems in number theory
![Page 2: Elementary problems in number theory - Univerzita Karlovaical/presentations/szabo.pdf · 2010-06-20 · Elementary problems in number theory Csaba Szab o E otv os Lor and University,](https://reader030.vdocuments.site/reader030/viewer/2022041102/5ede2d91ad6a402d66697b42/html5/thumbnails/2.jpg)
Problem 1.
How many numbers do you have to choose from 1 to 2n such that at leasttwo of them are relatively prime?
2n is enough (contains 1 and 2)
n is not enough: 2, 4, 6, . . . , 2n
n+1 is enough:
There are two consecutive numbers among them.Proof: Pigeon-holes: {1, 2}, {3, 4}, . . . , {2n − 1, 2n},
Csaba Szabo Elementary problems in number theory
![Page 3: Elementary problems in number theory - Univerzita Karlovaical/presentations/szabo.pdf · 2010-06-20 · Elementary problems in number theory Csaba Szab o E otv os Lor and University,](https://reader030.vdocuments.site/reader030/viewer/2022041102/5ede2d91ad6a402d66697b42/html5/thumbnails/3.jpg)
Problem 1.
How many numbers do you have to choose from 1 to 2n such that at leasttwo of them are relatively prime?
2n is enough (contains 1 and 2)
n is not enough: 2, 4, 6, . . . , 2n
n+1 is enough:
There are two consecutive numbers among them.Proof: Pigeon-holes: {1, 2}, {3, 4}, . . . , {2n − 1, 2n},
Csaba Szabo Elementary problems in number theory
![Page 4: Elementary problems in number theory - Univerzita Karlovaical/presentations/szabo.pdf · 2010-06-20 · Elementary problems in number theory Csaba Szab o E otv os Lor and University,](https://reader030.vdocuments.site/reader030/viewer/2022041102/5ede2d91ad6a402d66697b42/html5/thumbnails/4.jpg)
Problem 1.
How many numbers do you have to choose from 1 to 2n such that at leasttwo of them are relatively prime?
2n is enough (contains 1 and 2)
n is not enough: 2, 4, 6, . . . , 2n
n+1 is enough:
There are two consecutive numbers among them.Proof: Pigeon-holes: {1, 2}, {3, 4}, . . . , {2n − 1, 2n},
Csaba Szabo Elementary problems in number theory
![Page 5: Elementary problems in number theory - Univerzita Karlovaical/presentations/szabo.pdf · 2010-06-20 · Elementary problems in number theory Csaba Szab o E otv os Lor and University,](https://reader030.vdocuments.site/reader030/viewer/2022041102/5ede2d91ad6a402d66697b42/html5/thumbnails/5.jpg)
Problem 1.
How many numbers do you have to choose from 1 to 2n such that at leasttwo of them are relatively prime?
2n is enough (contains 1 and 2)
n is not enough: 2, 4, 6, . . . , 2n
n+1 is enough:
There are two consecutive numbers among them.Proof: Pigeon-holes: {1, 2}, {3, 4}, . . . , {2n − 1, 2n},
Csaba Szabo Elementary problems in number theory
![Page 6: Elementary problems in number theory - Univerzita Karlovaical/presentations/szabo.pdf · 2010-06-20 · Elementary problems in number theory Csaba Szab o E otv os Lor and University,](https://reader030.vdocuments.site/reader030/viewer/2022041102/5ede2d91ad6a402d66697b42/html5/thumbnails/6.jpg)
Problem 1.
How many numbers do you have to choose from 1 to 2n such that at leasttwo of them are relatively prime?
2n is enough (contains 1 and 2)
n is not enough: 2, 4, 6, . . . , 2n
n+1 is enough:
There are two consecutive numbers among them.Proof: Pigeon-holes: {1, 2}, {3, 4}, . . . , {2n − 1, 2n},
Csaba Szabo Elementary problems in number theory
![Page 7: Elementary problems in number theory - Univerzita Karlovaical/presentations/szabo.pdf · 2010-06-20 · Elementary problems in number theory Csaba Szab o E otv os Lor and University,](https://reader030.vdocuments.site/reader030/viewer/2022041102/5ede2d91ad6a402d66697b42/html5/thumbnails/7.jpg)
Problem 2.
How many numbers do you have to choose from 1 to 2n such that thereare two among them s.t. one divides the other?
2n is enough (contains 1 and 2)
n is not enough: n + 1, n + 2, . . . , 2n
n+1 is enough:
Proof: Pigeon-holes: {1 · 2t}, {3 · 2t}, . . . , {(2n − 1) · 2t}, labelled by oddnumbers.
Csaba Szabo Elementary problems in number theory
![Page 8: Elementary problems in number theory - Univerzita Karlovaical/presentations/szabo.pdf · 2010-06-20 · Elementary problems in number theory Csaba Szab o E otv os Lor and University,](https://reader030.vdocuments.site/reader030/viewer/2022041102/5ede2d91ad6a402d66697b42/html5/thumbnails/8.jpg)
Problem 2.
How many numbers do you have to choose from 1 to 2n such that thereare two among them s.t. one divides the other?
2n is enough (contains 1 and 2)
n is not enough: n + 1, n + 2, . . . , 2n
n+1 is enough:
Proof: Pigeon-holes: {1 · 2t}, {3 · 2t}, . . . , {(2n − 1) · 2t}, labelled by oddnumbers.
Csaba Szabo Elementary problems in number theory
![Page 9: Elementary problems in number theory - Univerzita Karlovaical/presentations/szabo.pdf · 2010-06-20 · Elementary problems in number theory Csaba Szab o E otv os Lor and University,](https://reader030.vdocuments.site/reader030/viewer/2022041102/5ede2d91ad6a402d66697b42/html5/thumbnails/9.jpg)
Problem 2.
How many numbers do you have to choose from 1 to 2n such that thereare two among them s.t. one divides the other?
2n is enough (contains 1 and 2)
n is not enough: n + 1, n + 2, . . . , 2n
n+1 is enough:
Proof: Pigeon-holes: {1 · 2t}, {3 · 2t}, . . . , {(2n − 1) · 2t}, labelled by oddnumbers.
Csaba Szabo Elementary problems in number theory
![Page 10: Elementary problems in number theory - Univerzita Karlovaical/presentations/szabo.pdf · 2010-06-20 · Elementary problems in number theory Csaba Szab o E otv os Lor and University,](https://reader030.vdocuments.site/reader030/viewer/2022041102/5ede2d91ad6a402d66697b42/html5/thumbnails/10.jpg)
Problem 2.
How many numbers do you have to choose from 1 to 2n such that thereare two among them s.t. one divides the other?
2n is enough (contains 1 and 2)
n is not enough: n + 1, n + 2, . . . , 2n
n+1 is enough:
Proof: Pigeon-holes: {1 · 2t}, {3 · 2t}, . . . , {(2n − 1) · 2t}, labelled by oddnumbers.
Csaba Szabo Elementary problems in number theory
![Page 11: Elementary problems in number theory - Univerzita Karlovaical/presentations/szabo.pdf · 2010-06-20 · Elementary problems in number theory Csaba Szab o E otv os Lor and University,](https://reader030.vdocuments.site/reader030/viewer/2022041102/5ede2d91ad6a402d66697b42/html5/thumbnails/11.jpg)
Problem 2.
How many numbers do you have to choose from 1 to 2n such that thereare two among them s.t. one divides the other?
2n is enough (contains 1 and 2)
n is not enough: n + 1, n + 2, . . . , 2n
n+1 is enough:
Proof: Pigeon-holes: {1 · 2t}, {3 · 2t}, . . . , {(2n − 1) · 2t}, labelled by oddnumbers.
Csaba Szabo Elementary problems in number theory
![Page 12: Elementary problems in number theory - Univerzita Karlovaical/presentations/szabo.pdf · 2010-06-20 · Elementary problems in number theory Csaba Szab o E otv os Lor and University,](https://reader030.vdocuments.site/reader030/viewer/2022041102/5ede2d91ad6a402d66697b42/html5/thumbnails/12.jpg)
Problem 3.
How many numbers do you have to choose such that the sum of a few ofthem is divisible by n?
n − 1 is not enough: 1, 1, . . . , 1
n is enough:
Proof: Pigeon-holes: residue classesPigeons: a1, a1 + a2, . . . , a1 + a2 + · · ·+ an
Two in the same pigeon-hole:
l∑1
ai −k∑1
ai =l∑k
ai
Csaba Szabo Elementary problems in number theory
![Page 13: Elementary problems in number theory - Univerzita Karlovaical/presentations/szabo.pdf · 2010-06-20 · Elementary problems in number theory Csaba Szab o E otv os Lor and University,](https://reader030.vdocuments.site/reader030/viewer/2022041102/5ede2d91ad6a402d66697b42/html5/thumbnails/13.jpg)
Problem 3.
How many numbers do you have to choose such that the sum of a few ofthem is divisible by n?
n − 1 is not enough: 1, 1, . . . , 1
n is enough:
Proof: Pigeon-holes: residue classesPigeons: a1, a1 + a2, . . . , a1 + a2 + · · ·+ an
Two in the same pigeon-hole:
l∑1
ai −k∑1
ai =l∑k
ai
Csaba Szabo Elementary problems in number theory
![Page 14: Elementary problems in number theory - Univerzita Karlovaical/presentations/szabo.pdf · 2010-06-20 · Elementary problems in number theory Csaba Szab o E otv os Lor and University,](https://reader030.vdocuments.site/reader030/viewer/2022041102/5ede2d91ad6a402d66697b42/html5/thumbnails/14.jpg)
Problem 3.
How many numbers do you have to choose such that the sum of a few ofthem is divisible by n?
n − 1 is not enough: 1, 1, . . . , 1
n is enough:
Proof: Pigeon-holes: residue classesPigeons: a1, a1 + a2, . . . , a1 + a2 + · · ·+ an
Two in the same pigeon-hole:
l∑1
ai −k∑1
ai =l∑k
ai
Csaba Szabo Elementary problems in number theory
![Page 15: Elementary problems in number theory - Univerzita Karlovaical/presentations/szabo.pdf · 2010-06-20 · Elementary problems in number theory Csaba Szab o E otv os Lor and University,](https://reader030.vdocuments.site/reader030/viewer/2022041102/5ede2d91ad6a402d66697b42/html5/thumbnails/15.jpg)
Problem 3.
How many numbers do you have to choose such that the sum of a few ofthem is divisible by n?
n − 1 is not enough: 1, 1, . . . , 1
n is enough:
Proof: Pigeon-holes: residue classesPigeons: a1, a1 + a2, . . . , a1 + a2 + · · ·+ an
Two in the same pigeon-hole:
l∑1
ai −k∑1
ai =l∑k
ai
Csaba Szabo Elementary problems in number theory
![Page 16: Elementary problems in number theory - Univerzita Karlovaical/presentations/szabo.pdf · 2010-06-20 · Elementary problems in number theory Csaba Szab o E otv os Lor and University,](https://reader030.vdocuments.site/reader030/viewer/2022041102/5ede2d91ad6a402d66697b42/html5/thumbnails/16.jpg)
Problem 4.
How many numbers do you have to choose such that the sum of n ofthem is divisible by n?
2n − 2 is not enough: 0, 0, . . . , 0, 1, 1, . . . , 1
n2 − n + 1 is enough:
Proof: Pigeon-holes: resiude classesAt least n in a single pigeon-hole
(n − 1)2 + 1 is enough:
Is there a better bound?
Csaba Szabo Elementary problems in number theory
![Page 17: Elementary problems in number theory - Univerzita Karlovaical/presentations/szabo.pdf · 2010-06-20 · Elementary problems in number theory Csaba Szab o E otv os Lor and University,](https://reader030.vdocuments.site/reader030/viewer/2022041102/5ede2d91ad6a402d66697b42/html5/thumbnails/17.jpg)
Problem 4.
How many numbers do you have to choose such that the sum of n ofthem is divisible by n?
2n − 2 is not enough: 0, 0, . . . , 0, 1, 1, . . . , 1
n2 − n + 1 is enough:
Proof: Pigeon-holes: resiude classesAt least n in a single pigeon-hole
(n − 1)2 + 1 is enough:
Is there a better bound?
Csaba Szabo Elementary problems in number theory
![Page 18: Elementary problems in number theory - Univerzita Karlovaical/presentations/szabo.pdf · 2010-06-20 · Elementary problems in number theory Csaba Szab o E otv os Lor and University,](https://reader030.vdocuments.site/reader030/viewer/2022041102/5ede2d91ad6a402d66697b42/html5/thumbnails/18.jpg)
Problem 4.
How many numbers do you have to choose such that the sum of n ofthem is divisible by n?
2n − 2 is not enough: 0, 0, . . . , 0, 1, 1, . . . , 1
n2 − n + 1 is enough:
Proof: Pigeon-holes: resiude classesAt least n in a single pigeon-hole
(n − 1)2 + 1 is enough:
Is there a better bound?
Csaba Szabo Elementary problems in number theory
![Page 19: Elementary problems in number theory - Univerzita Karlovaical/presentations/szabo.pdf · 2010-06-20 · Elementary problems in number theory Csaba Szab o E otv os Lor and University,](https://reader030.vdocuments.site/reader030/viewer/2022041102/5ede2d91ad6a402d66697b42/html5/thumbnails/19.jpg)
Problem 4.
How many numbers do you have to choose such that the sum of n ofthem is divisible by n?
2n − 2 is not enough: 0, 0, . . . , 0, 1, 1, . . . , 1
n2 − n + 1 is enough:
Proof: Pigeon-holes: resiude classesAt least n in a single pigeon-hole
(n − 1)2 + 1 is enough:
Is there a better bound?
Csaba Szabo Elementary problems in number theory
![Page 20: Elementary problems in number theory - Univerzita Karlovaical/presentations/szabo.pdf · 2010-06-20 · Elementary problems in number theory Csaba Szab o E otv os Lor and University,](https://reader030.vdocuments.site/reader030/viewer/2022041102/5ede2d91ad6a402d66697b42/html5/thumbnails/20.jpg)
Problem 4.
How many numbers do you have to choose such that the sum of n ofthem is divisible by n?
2n − 2 is not enough: 0, 0, . . . , 0, 1, 1, . . . , 1
n2 − n + 1 is enough:
Proof: Pigeon-holes: resiude classesAt least n in a single pigeon-hole
(n − 1)2 + 1 is enough:
Is there a better bound?
Csaba Szabo Elementary problems in number theory
![Page 21: Elementary problems in number theory - Univerzita Karlovaical/presentations/szabo.pdf · 2010-06-20 · Elementary problems in number theory Csaba Szab o E otv os Lor and University,](https://reader030.vdocuments.site/reader030/viewer/2022041102/5ede2d91ad6a402d66697b42/html5/thumbnails/21.jpg)
Chevalley’s Theorem
Lemma
Let A1, . . . ,An be subsets of Fp, the p-element field, andf ∈ Fp[x1, . . . , xn] such that
n∑i=1
(|Ai | − 1) > (p − 1) deg f .
If the set {a ∈ A1 × · · · × An|f (a) = 0} is not empty, then it has at leasttwo different elements.
Csaba Szabo Elementary problems in number theory
![Page 22: Elementary problems in number theory - Univerzita Karlovaical/presentations/szabo.pdf · 2010-06-20 · Elementary problems in number theory Csaba Szab o E otv os Lor and University,](https://reader030.vdocuments.site/reader030/viewer/2022041102/5ede2d91ad6a402d66697b42/html5/thumbnails/22.jpg)
Chevalley’s Theorem
Lemma
Let A1, . . . ,An be subsets of Fp, the p-element field, andf ∈ Fp[x1, . . . , xn] such that
n∑i=1
(|Ai | − 1) > (p − 1) deg f .
If the set {a ∈ A1 × · · · × An|f (a) = 0} is not empty, then it has at leasttwo different elements.
Csaba Szabo Elementary problems in number theory
![Page 23: Elementary problems in number theory - Univerzita Karlovaical/presentations/szabo.pdf · 2010-06-20 · Elementary problems in number theory Csaba Szab o E otv os Lor and University,](https://reader030.vdocuments.site/reader030/viewer/2022041102/5ede2d91ad6a402d66697b42/html5/thumbnails/23.jpg)
Csaba Szabo Elementary problems in number theory
![Page 24: Elementary problems in number theory - Univerzita Karlovaical/presentations/szabo.pdf · 2010-06-20 · Elementary problems in number theory Csaba Szab o E otv os Lor and University,](https://reader030.vdocuments.site/reader030/viewer/2022041102/5ede2d91ad6a402d66697b42/html5/thumbnails/24.jpg)
Why am I talking about these problems?
Csaba Szabo Elementary problems in number theory
![Page 25: Elementary problems in number theory - Univerzita Karlovaical/presentations/szabo.pdf · 2010-06-20 · Elementary problems in number theory Csaba Szab o E otv os Lor and University,](https://reader030.vdocuments.site/reader030/viewer/2022041102/5ede2d91ad6a402d66697b42/html5/thumbnails/25.jpg)
Definition
Let A be an algebra and t1 and t2 be two terms over A.
We say that t1 and t2 are equivalent over A if t1(a) = t2(a) for everysubstitution a ∈ A
Definition
ID-CHECK A
Let A be an algebra
Input: t1 and t2 two terms over A
Question: Are t1 and t2 equivalent over A?
Always decidable: check every substitution
Csaba Szabo Elementary problems in number theory
![Page 26: Elementary problems in number theory - Univerzita Karlovaical/presentations/szabo.pdf · 2010-06-20 · Elementary problems in number theory Csaba Szab o E otv os Lor and University,](https://reader030.vdocuments.site/reader030/viewer/2022041102/5ede2d91ad6a402d66697b42/html5/thumbnails/26.jpg)
Definition
Let A be an algebra and t1 and t2 be two terms over A.
We say that t1 and t2 are equivalent over A if t1(a) = t2(a) for everysubstitution a ∈ A
Definition
ID-CHECK A
Let A be an algebra
Input: t1 and t2 two terms over A
Question: Are t1 and t2 equivalent over A?
Always decidable: check every substitution
Csaba Szabo Elementary problems in number theory
![Page 27: Elementary problems in number theory - Univerzita Karlovaical/presentations/szabo.pdf · 2010-06-20 · Elementary problems in number theory Csaba Szab o E otv os Lor and University,](https://reader030.vdocuments.site/reader030/viewer/2022041102/5ede2d91ad6a402d66697b42/html5/thumbnails/27.jpg)
Definition
Let A be an algebra and t1 and t2 be two terms over A.
We say that t1 and t2 are equivalent over A if t1(a) = t2(a) for everysubstitution a ∈ A
Definition
ID-CHECK A
Let A be an algebra
Input: t1 and t2 two terms over A
Question: Are t1 and t2 equivalent over A?
Always decidable: check every substitution
Csaba Szabo Elementary problems in number theory
![Page 28: Elementary problems in number theory - Univerzita Karlovaical/presentations/szabo.pdf · 2010-06-20 · Elementary problems in number theory Csaba Szab o E otv os Lor and University,](https://reader030.vdocuments.site/reader030/viewer/2022041102/5ede2d91ad6a402d66697b42/html5/thumbnails/28.jpg)
Definition
Let A be an algebra and t1 and t2 be two terms over A.
We say that t1 and t2 are equivalent over A if t1(a) = t2(a) for everysubstitution a ∈ A
Definition
ID-CHECK A
Let A be an algebra
Input: t1 and t2 two terms over A
Question: Are t1 and t2 equivalent over A?
Always decidable: check every substitution
Csaba Szabo Elementary problems in number theory
![Page 29: Elementary problems in number theory - Univerzita Karlovaical/presentations/szabo.pdf · 2010-06-20 · Elementary problems in number theory Csaba Szab o E otv os Lor and University,](https://reader030.vdocuments.site/reader030/viewer/2022041102/5ede2d91ad6a402d66697b42/html5/thumbnails/29.jpg)
Definition
Let A be an algebra and t1 and t2 be two terms over A.
We say that t1 and t2 are equivalent over A if t1(a) = t2(a) for everysubstitution a ∈ A
Definition
ID-CHECK A
Let A be an algebra
Input: t1 and t2 two terms over A
Question: Are t1 and t2 equivalent over A?
Always decidable: check every substitution
Csaba Szabo Elementary problems in number theory
![Page 30: Elementary problems in number theory - Univerzita Karlovaical/presentations/szabo.pdf · 2010-06-20 · Elementary problems in number theory Csaba Szab o E otv os Lor and University,](https://reader030.vdocuments.site/reader030/viewer/2022041102/5ede2d91ad6a402d66697b42/html5/thumbnails/30.jpg)
Definition
Let A be an algebra and t1 and t2 be two terms over A.
We say that t1 and t2 are equivalent over A if t1(a) = t2(a) for everysubstitution a ∈ A
Definition
ID-CHECK A
Let A be an algebra
Input: t1 and t2 two terms over A
Question: Are t1 and t2 equivalent over A?
Always decidable: check every substitution
Csaba Szabo Elementary problems in number theory
![Page 31: Elementary problems in number theory - Univerzita Karlovaical/presentations/szabo.pdf · 2010-06-20 · Elementary problems in number theory Csaba Szab o E otv os Lor and University,](https://reader030.vdocuments.site/reader030/viewer/2022041102/5ede2d91ad6a402d66697b42/html5/thumbnails/31.jpg)
Definition
Let A be an algebra and t1 and t2 be two terms over A.
We say that t1 and t2 are equivalent over A if t1(a) = t2(a) for everysubstitution a ∈ A
Definition
ID-CHECK A
Let A be an algebra
Input: t1 and t2 two terms over A
Question: Are t1 and t2 equivalent over A?
Always decidable: check every substitution
Csaba Szabo Elementary problems in number theory
![Page 32: Elementary problems in number theory - Univerzita Karlovaical/presentations/szabo.pdf · 2010-06-20 · Elementary problems in number theory Csaba Szab o E otv os Lor and University,](https://reader030.vdocuments.site/reader030/viewer/2022041102/5ede2d91ad6a402d66697b42/html5/thumbnails/32.jpg)
Another question
t1 and t2 two polynomials over A
POL-SAT : Does t1 = t2 have a solution?
Rings
ID-CHECK R: Is t = t1 − t2 identically 0?
POL-SAT R: Does t = t1 − t2 have a root?
Csaba Szabo Elementary problems in number theory
![Page 33: Elementary problems in number theory - Univerzita Karlovaical/presentations/szabo.pdf · 2010-06-20 · Elementary problems in number theory Csaba Szab o E otv os Lor and University,](https://reader030.vdocuments.site/reader030/viewer/2022041102/5ede2d91ad6a402d66697b42/html5/thumbnails/33.jpg)
Another question
t1 and t2 two polynomials over A
POL-SAT : Does t1 = t2 have a solution?
Rings
ID-CHECK R: Is t = t1 − t2 identically 0?
POL-SAT R: Does t = t1 − t2 have a root?
Csaba Szabo Elementary problems in number theory
![Page 34: Elementary problems in number theory - Univerzita Karlovaical/presentations/szabo.pdf · 2010-06-20 · Elementary problems in number theory Csaba Szab o E otv os Lor and University,](https://reader030.vdocuments.site/reader030/viewer/2022041102/5ede2d91ad6a402d66697b42/html5/thumbnails/34.jpg)
Another question
t1 and t2 two polynomials over A
POL-SAT : Does t1 = t2 have a solution?
Rings
ID-CHECK R: Is t = t1 − t2 identically 0?
POL-SAT R: Does t = t1 − t2 have a root?
Csaba Szabo Elementary problems in number theory
![Page 35: Elementary problems in number theory - Univerzita Karlovaical/presentations/szabo.pdf · 2010-06-20 · Elementary problems in number theory Csaba Szab o E otv os Lor and University,](https://reader030.vdocuments.site/reader030/viewer/2022041102/5ede2d91ad6a402d66697b42/html5/thumbnails/35.jpg)
Another question
t1 and t2 two polynomials over A
POL-SAT : Does t1 = t2 have a solution?
Rings
ID-CHECK R: Is t = t1 − t2 identically 0?
POL-SAT R: Does t = t1 − t2 have a root?
Csaba Szabo Elementary problems in number theory
![Page 36: Elementary problems in number theory - Univerzita Karlovaical/presentations/szabo.pdf · 2010-06-20 · Elementary problems in number theory Csaba Szab o E otv os Lor and University,](https://reader030.vdocuments.site/reader030/viewer/2022041102/5ede2d91ad6a402d66697b42/html5/thumbnails/36.jpg)
Another question
t1 and t2 two polynomials over A
POL-SAT : Does t1 = t2 have a solution?
Rings
ID-CHECK R: Is t = t1 − t2 identically 0?
POL-SAT R: Does t = t1 − t2 have a root?
Csaba Szabo Elementary problems in number theory
![Page 37: Elementary problems in number theory - Univerzita Karlovaical/presentations/szabo.pdf · 2010-06-20 · Elementary problems in number theory Csaba Szab o E otv os Lor and University,](https://reader030.vdocuments.site/reader030/viewer/2022041102/5ede2d91ad6a402d66697b42/html5/thumbnails/37.jpg)
Another question
t1 and t2 two polynomials over A
POL-SAT : Does t1 = t2 have a solution?
Rings
ID-CHECK R: Is t = t1 − t2 identically 0?
POL-SAT R: Does t = t1 − t2 have a root?
Csaba Szabo Elementary problems in number theory
![Page 38: Elementary problems in number theory - Univerzita Karlovaical/presentations/szabo.pdf · 2010-06-20 · Elementary problems in number theory Csaba Szab o E otv os Lor and University,](https://reader030.vdocuments.site/reader030/viewer/2022041102/5ede2d91ad6a402d66697b42/html5/thumbnails/38.jpg)
Abelian groups
A Abelian group
t(x1, . . . , xn) = xk11 . . . xkn
n
t(x1, . . . , xn)?≡ 1 over A
xk11 . . . xkn
n ≡ 1
∀i 6= m xi = 1 =⇒ xkmm ≡ 1
exp A | km for every m
xk11 . . . xkn
n ≡ 1⇐⇒ ∀m : exp A | km
Csaba Szabo Elementary problems in number theory
![Page 39: Elementary problems in number theory - Univerzita Karlovaical/presentations/szabo.pdf · 2010-06-20 · Elementary problems in number theory Csaba Szab o E otv os Lor and University,](https://reader030.vdocuments.site/reader030/viewer/2022041102/5ede2d91ad6a402d66697b42/html5/thumbnails/39.jpg)
Abelian groups
A Abelian group
t(x1, . . . , xn) = xk11 . . . xkn
n
t(x1, . . . , xn)?≡ 1 over A
xk11 . . . xkn
n ≡ 1
∀i 6= m xi = 1 =⇒ xkmm ≡ 1
exp A | km for every m
xk11 . . . xkn
n ≡ 1⇐⇒ ∀m : exp A | km
Csaba Szabo Elementary problems in number theory
![Page 40: Elementary problems in number theory - Univerzita Karlovaical/presentations/szabo.pdf · 2010-06-20 · Elementary problems in number theory Csaba Szab o E otv os Lor and University,](https://reader030.vdocuments.site/reader030/viewer/2022041102/5ede2d91ad6a402d66697b42/html5/thumbnails/40.jpg)
Abelian groups
A Abelian group
t(x1, . . . , xn) = xk11 . . . xkn
n
t(x1, . . . , xn)?≡ 1 over A
xk11 . . . xkn
n ≡ 1
∀i 6= m xi = 1 =⇒ xkmm ≡ 1
exp A | km for every m
xk11 . . . xkn
n ≡ 1⇐⇒ ∀m : exp A | km
Csaba Szabo Elementary problems in number theory
![Page 41: Elementary problems in number theory - Univerzita Karlovaical/presentations/szabo.pdf · 2010-06-20 · Elementary problems in number theory Csaba Szab o E otv os Lor and University,](https://reader030.vdocuments.site/reader030/viewer/2022041102/5ede2d91ad6a402d66697b42/html5/thumbnails/41.jpg)
Abelian groups
A Abelian group
t(x1, . . . , xn) = xk11 . . . xkn
n
t(x1, . . . , xn)?≡ 1 over A
xk11 . . . xkn
n ≡ 1
∀i 6= m xi = 1 =⇒ xkmm ≡ 1
exp A | km for every m
xk11 . . . xkn
n ≡ 1⇐⇒ ∀m : exp A | km
Csaba Szabo Elementary problems in number theory
![Page 42: Elementary problems in number theory - Univerzita Karlovaical/presentations/szabo.pdf · 2010-06-20 · Elementary problems in number theory Csaba Szab o E otv os Lor and University,](https://reader030.vdocuments.site/reader030/viewer/2022041102/5ede2d91ad6a402d66697b42/html5/thumbnails/42.jpg)
Abelian groups
A Abelian group
t(x1, . . . , xn) = xk11 . . . xkn
n
t(x1, . . . , xn)?≡ 1 over A
xk11 . . . xkn
n ≡ 1
∀i 6= m xi = 1 =⇒ xkmm ≡ 1
exp A | km for every m
xk11 . . . xkn
n ≡ 1⇐⇒ ∀m : exp A | km
Csaba Szabo Elementary problems in number theory
![Page 43: Elementary problems in number theory - Univerzita Karlovaical/presentations/szabo.pdf · 2010-06-20 · Elementary problems in number theory Csaba Szab o E otv os Lor and University,](https://reader030.vdocuments.site/reader030/viewer/2022041102/5ede2d91ad6a402d66697b42/html5/thumbnails/43.jpg)
Abelian groups
A Abelian group
t(x1, . . . , xn) = xk11 . . . xkn
n
t(x1, . . . , xn)?≡ 1 over A
xk11 . . . xkn
n ≡ 1
∀i 6= m xi = 1 =⇒ xkmm ≡ 1
exp A | km for every m
xk11 . . . xkn
n ≡ 1⇐⇒ ∀m : exp A | km
Csaba Szabo Elementary problems in number theory
![Page 44: Elementary problems in number theory - Univerzita Karlovaical/presentations/szabo.pdf · 2010-06-20 · Elementary problems in number theory Csaba Szab o E otv os Lor and University,](https://reader030.vdocuments.site/reader030/viewer/2022041102/5ede2d91ad6a402d66697b42/html5/thumbnails/44.jpg)
Abelian groups
A Abelian group
t(x1, . . . , xn) = xk11 . . . xkn
n
t(x1, . . . , xn)?≡ 1 over A
xk11 . . . xkn
n ≡ 1
∀i 6= m xi = 1 =⇒ xkmm ≡ 1
exp A | km for every m
xk11 . . . xkn
n ≡ 1⇐⇒ ∀m : exp A | km
Csaba Szabo Elementary problems in number theory
![Page 45: Elementary problems in number theory - Univerzita Karlovaical/presentations/szabo.pdf · 2010-06-20 · Elementary problems in number theory Csaba Szab o E otv os Lor and University,](https://reader030.vdocuments.site/reader030/viewer/2022041102/5ede2d91ad6a402d66697b42/html5/thumbnails/45.jpg)
Nilpotent rings
Idziak- Szabo
Let A be a nilpotent algebra of size r and of nilpotency class k, andf (x) ∈ R[x1, x2, . . . , xn] be a polynomial over A.Then for every a ∈ Rn there is a b ∈ Rn such that
bi = 0 or bi = ai
bi = ai for at most rr...rk
many i-s (there are k-many r -s in the tower)
f (a) = f (b)
G. Horvath
same bound, simpler proof for groups and rings
Csaba Szabo Elementary problems in number theory
![Page 46: Elementary problems in number theory - Univerzita Karlovaical/presentations/szabo.pdf · 2010-06-20 · Elementary problems in number theory Csaba Szab o E otv os Lor and University,](https://reader030.vdocuments.site/reader030/viewer/2022041102/5ede2d91ad6a402d66697b42/html5/thumbnails/46.jpg)
Nilpotent rings
Idziak- Szabo
Let A be a nilpotent algebra of size r and of nilpotency class k, andf (x) ∈ R[x1, x2, . . . , xn] be a polynomial over A.Then for every a ∈ Rn there is a b ∈ Rn such that
bi = 0 or bi = ai
bi = ai for at most rr...rk
many i-s (there are k-many r -s in the tower)
f (a) = f (b)
G. Horvath
same bound, simpler proof for groups and rings
Csaba Szabo Elementary problems in number theory
![Page 47: Elementary problems in number theory - Univerzita Karlovaical/presentations/szabo.pdf · 2010-06-20 · Elementary problems in number theory Csaba Szab o E otv os Lor and University,](https://reader030.vdocuments.site/reader030/viewer/2022041102/5ede2d91ad6a402d66697b42/html5/thumbnails/47.jpg)
Nilpotent rings
Idziak- Szabo
Let A be a nilpotent algebra of size r and of nilpotency class k, andf (x) ∈ R[x1, x2, . . . , xn] be a polynomial over A.Then for every a ∈ Rn there is a b ∈ Rn such that
bi = 0 or bi = ai
bi = ai for at most rr...rk
many i-s (there are k-many r -s in the tower)
f (a) = f (b)
G. Horvath
same bound, simpler proof for groups and rings
Csaba Szabo Elementary problems in number theory
![Page 48: Elementary problems in number theory - Univerzita Karlovaical/presentations/szabo.pdf · 2010-06-20 · Elementary problems in number theory Csaba Szab o E otv os Lor and University,](https://reader030.vdocuments.site/reader030/viewer/2022041102/5ede2d91ad6a402d66697b42/html5/thumbnails/48.jpg)
Nilpotent rings
Idziak- Szabo
Let A be a nilpotent algebra of size r and of nilpotency class k, andf (x) ∈ R[x1, x2, . . . , xn] be a polynomial over A.Then for every a ∈ Rn there is a b ∈ Rn such that
bi = 0 or bi = ai
bi = ai for at most rr...rk
many i-s (there are k-many r -s in the tower)
f (a) = f (b)
G. Horvath
same bound, simpler proof for groups and rings
Csaba Szabo Elementary problems in number theory
![Page 49: Elementary problems in number theory - Univerzita Karlovaical/presentations/szabo.pdf · 2010-06-20 · Elementary problems in number theory Csaba Szab o E otv os Lor and University,](https://reader030.vdocuments.site/reader030/viewer/2022041102/5ede2d91ad6a402d66697b42/html5/thumbnails/49.jpg)
Nilpotent rings
Idziak- Szabo
Let A be a nilpotent algebra of size r and of nilpotency class k, andf (x) ∈ R[x1, x2, . . . , xn] be a polynomial over A.Then for every a ∈ Rn there is a b ∈ Rn such that
bi = 0 or bi = ai
bi = ai for at most rr...rk
many i-s (there are k-many r -s in the tower)
f (a) = f (b)
G. Horvath
same bound, simpler proof for groups and rings
Csaba Szabo Elementary problems in number theory
![Page 50: Elementary problems in number theory - Univerzita Karlovaical/presentations/szabo.pdf · 2010-06-20 · Elementary problems in number theory Csaba Szab o E otv os Lor and University,](https://reader030.vdocuments.site/reader030/viewer/2022041102/5ede2d91ad6a402d66697b42/html5/thumbnails/50.jpg)
Nilpotent rings
Let F (a) = F (a1, . . . , an) = b.
For H ⊆ {1, 2, . . . , n} let aH =
{ai if i ∈ H
0 if i /∈ H
ϕ(H) = see board
ϕ(H) =∑X⊆H
ϕ(X )
f (x) =∑H
ϕ(H)∏i∈H
xi
Clearly , ϕ(H) = F (aH)
Csaba Szabo Elementary problems in number theory
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recall
ϕ(H) =∑
X⊆H
ϕ(X ) and f (x) =∑H
ϕ(H)∏i∈H
xi
f (1) =∑
ϕ(X ) =ϕ({1, . . . , n}) = F (a)
f (χ(H)) =∑X⊆H
ϕ(X )=ϕ(H) = F (aH)
g(x) = f (x)− f (1)g(1) = 0g(χ(H)) = 0 ⇐⇒ F (aH) = b
Csaba Szabo Elementary problems in number theory
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recall
ϕ(H) =∑
X⊆H
ϕ(X ) and f (x) =∑H
ϕ(H)∏i∈H
xi
f (1) =∑
ϕ(X ) =ϕ({1, . . . , n}) = F (a)
f (χ(H)) =∑X⊆H
ϕ(X )=ϕ(H) = F (aH)
g(x) = f (x)− f (1)g(1) = 0g(χ(H)) = 0 ⇐⇒ F (aH) = b
Csaba Szabo Elementary problems in number theory
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recall
ϕ(H) =∑
X⊆H
ϕ(X ) and f (x) =∑H
ϕ(H)∏i∈H
xi
f (1) =∑
ϕ(X ) =ϕ({1, . . . , n}) = F (a)
f (χ(H)) =∑X⊆H
ϕ(X )=ϕ(H) = F (aH)
g(x) = f (x)− f (1)g(1) = 0g(χ(H)) = 0 ⇐⇒ F (aH) = b
Csaba Szabo Elementary problems in number theory
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Chevalley’s Theorem, again
Recall
Let A1, . . . ,An be subsets of Fp, the p-element field, andf ∈ Fp[x1, . . . , xn] such that
n∑i=1
(|Ai | − 1) > (p − 1) deg f .
If the set {a ∈ A1 × · · · × An|f (a) = 0} is not empty, then it has at leasttwo different elements.
Apply Lemma for g(x) and Ai = {0, 1}. g(1) = 0.If n > k(p − 1), there is an other root.
Csaba Szabo Elementary problems in number theory
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Chevalley’s Theorem, again
Recall
Let A1, . . . ,An be subsets of Fp, the p-element field, andf ∈ Fp[x1, . . . , xn] such that
n∑i=1
(|Ai | − 1) > (p − 1) deg f .
If the set {a ∈ A1 × · · · × An|f (a) = 0} is not empty, then it has at leasttwo different elements.
Apply Lemma for g(x) and Ai = {0, 1}. g(1) = 0.If n > k(p − 1), there is an other root.
Csaba Szabo Elementary problems in number theory