Pr oblemen

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Problemen NAW 5/15 nr. 1 maart 2014

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Pr oblemen

Redactie:

Johan Bosman Gabriele Dalla Torre Christophe Debry Jinbi Jin Marco Streng Wouter Zomervrucht Problemenrubriek NAW Mathematisch Instituut Universiteit Leiden Postbus 9512 2300 RA Leiden

problems@nieuwarchief.nl www.nieuwarchief.nl/problems

This Problem Section is open to everyone; everybody is encouraged to send in solutions and propose problems. Group contributions are welcome.

For each problem, the most elegant correct solution will be rewarded with a book token worth D20. At times there will be a Star Problem, to which the proposer does not know any solution.

For the first correct solution sent in within one year there is a prize of^D100.

When proposing a problem, please either include a complete solution or indicate that it is intended as a Star Problem. Electronic submissions of problems and solutions are preferred (problems@nieuwarchief.nl).

The deadline for solutions to the problems in this edition is 1 June 2014.

Problem A(proposed by Hendrik Lenstra)

LetGbe a group, and leta, b ∈ Gbe two elements satisfying{gag⁻¹:g ∈ G} = {a, b}. Prove that for allc ∈ Gone hasabc = cba.

Problem B(the attribution will appear in the September issue of 2014)

LetKbe a field, and consider for all positive integersnthe subsetSnofx ∈ K^∗that can be written as the sum ofnsquares inK. Show that the subgroup ofK^∗generated byS_nis equal to St(n). Here, for a positive integern, we denote byt(n)the smallest power of two that is greater than or equal ton.

Problem C(folklore)

Given five pairwise distinct pointsA, B, C, D, Ein the plane, no three of which are collinear, and given a linelin the plane not passing through any of the five points. Assume thatlintersects the conic sectioncpassing throughA, B, C, D, E. Construct the intersection points oflandc. One of the solutions which uses the smallest number of moves will be awarded the book token.

Here, given a collection of points, lines, and circles in the plane, a move consists of adding to the collection either a line through two of the points, or a circle centered at one of them and passing through another. At any time one is allowed to freely add any intersection point among the lines and circles, as well as any sufficiently general point, either in the plane, or on any of the lines or circles.

For example, given a lineℓand a pointPonℓone can construct a line throughPand perpen- dicular toℓin three moves as follows. Choose a pointMnot onℓ. For the first move, take the circleCcentered atMand going throughP. LetQbe the second point of intersection between Candℓ. For the second move, add the line throughQandMand letRbe the second point of intersection between this line andC. Finally, add the lineP R, which is perpendicular toℓ.

Edition 2013-3 We received solutions from Pieter de Groen (Brussels), Alex Heinis (Amsterdam), Tejaswi Navilarekallu and Sander Zwegers (Keulen).

Problem 2013-3/A (proposed by Hendrik Lenstra)

Leta, b ∈ C. Show that if there exists an irreducible polynomialf ∈ Q[X]such thatf (a) = f (a + b) = f (a + 2b) = 0, thenb = 0.

Solution We received solutions from Alex Heinis, Tejaswi Navilarekallu and Sander Zwegers.

The following solution is based on that of Alex Heinis, who will receive the book token.

LetSbe the (finite) set of complex zeros off and letHbe the convex hull ofS, which is a polygon or a line segment. Letv ∈ Sbe a vertex ofH.

Sincefis irreducible, the Galois groupGof its splitting field acts transitively onS, hence there is an elementσ ∈ Gwithσ (a + b) = v.

Nowσ (a)andσ (a + 2b)are also inS, andv = σ (a + b)lies on the line segment connecting them. This contradicts the fact thatvis a vertex, unlessb = 0.

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Problem 2013-3/B (a result due to B. Konstant and N. Wallach [1])

Letnbe a positive integer, and lete_ij be an integer for all1 ≤j ≤ i ≤ n. Show that there exists ann × n–matrix with entries inZsuch that the eigenvalues of the top lefti × i–minor areei1, . . . , eii(with multiplicity).

SolutionWe received correct solutions from Pieter de Groen, Alex Heinis and Sander Zwegers.

All gave a solution similar to the one we present below. The book token goes to Sander Zwegers.

For any square matrixAwe denote the top lefti × i–minor bym_i(A). We also use the notation χA(λ) := det(λI − A)for the characteristic polynomial ofA. LetSnbe the set of alln × n– matricesA = (a_ij)over the polynomial ringZ[λ]such thata_i+1,i= 1for all^{1 ≤}i < nand such thata_i,j= 0ifi ≥ j + 2, i.e.,Ahas ones on the line just below its main diagonal and has zeroes below this line of ones. (We allowλin the matrices because we will work with characteristic polynomials.) It suffices to prove the following claim:

Claim. Letnbe a positive integer and leteijbe an integer for all1 ≤j ≤ i ≤ n. Then there exists a matrixA ∈ S_nwith entries inZsuch thatχ(m_i(A)) = (λ−e_i1) · · · (λ−e_ii)for all1 ≤i ≤ n.

We prove this claim by induction onn. The base casen = 1is trivial: the matrix(e11)will do. Now suppose that the claim is true for somen = N − 1 ≥ 1and let us prove it forn = N. So leteijbe an integer for all^{1 ≤}j ≤ i ≤ Nand letA ∈ S_N−1be the matrix we get from the claim forn = N −1 and the integerse_ijwith1 ≤j ≤ i ≤ N − 1. Since1, χ(m1(A)), . . . , χ(mN−1(A)), λχ(mN−1(A)) are monic polynomials overZof degree0, 1, . . . , N − 1, N(respectively), they form aZ–basis for the abelian group of polynomials of degree≤Nwith integer coefficients. So there exist integersx1, . . . , xN+1such that the degree–Npolynomial(λ − eN1) · · · (λ − eNN)is equal to the linear combination

−x1+ (−1)^N+1x_N+1λχ(m_N−1(A)) +

N

X

i =2

(−1)ⁱx_iχ(m_i−1(A)).

Since this polynomial is monic of degreeN, we know thatxN+1= (−1)^N+1. We claim that we can take the following matrix to prove the claim forn = N:

B =















x1

A x2

... 0 · · · 1 xN













 .

First of all,A ∈ SN−1implies thatB ∈ SN. Moreover, fori < Nwe haveχ(mi(B)) = χ(mi(A)) = (λ − e_i1) · · · (λ − e_ii)by the construction ofBand the choice ofA. So we are left to compute χ(mN(B)) = χ(B).

Lemma.For everyM = (M_ij) ∈S_n(n ≥ 2) we have

det(M) = (−1)ⁿ



−M1,n+

n

X

i=2

(−1)ⁱM_i,ndet(m_i−1(M))



.

Proof. Expand the determinant with respect to the last row to find that

det(M) = Mn,ndet(mn−1(M)) − det















M1,n

mn−2(M) M2,n

... 0 · · · 1 Mn−1,n













 .

Now use induction and the fact thatmi(mj(M)) = mi(M)for alli ≤ j ≤ n.

The lemma enables us to compute the characteristic polynomial ofB:

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Problemen NAW 5/15 nr. 1 maart 2014

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Oplossingen

χ(B) = (−1)^Ndet(B − λI) = −x1+

N

X

i=2

(−1)ⁱ(B − λI)_i,Ndet(m_i−1(B − λI))

= −x1+x2χ(m1(A)) + · · · + (−1)^N−1xN−1χ(mN−2(A)) + (−1)^N(x_N−λ)χ(m_N−1(A)).

By definition ofx1, . . . , x_N, this polynomial is equal to(λ−e_N1) · · · (λ−e_NN)and this concludes the proof of the claim forn = N.

Problem 2013-3/C (proposed by Bart de Smit and Hendrik Lenstra)

LetAbe a finite commutative unital ring. Does there exist a pair(B, f )withBa finite commutative unital ring in which every ideal is principal, andfan injective ring homomorphismA → B?

Solution We received no correct solutions. The answer to the question is “no”. LetAbe the finite commutative unital ringF2[x, y]/(x², y²). LetBbe a finite commutative unital ring in which every ideal is principal, and letf : A → Bbe a ring homomorphism. It now suffices to show thatfcannot be injective.

First note that the ring homomorphismf forces2 = 0inB. By assumption there exists an elementz ∈ Bsuch that we have the identity^f (x), f (y)

= (z)as ideals ofB. Hence there existb, c ∈ Bsuch thatf (x) = bzandf (y) = cz, sof (xy) = bcz². Writingz = df (x) + ef (y) withd, e ∈ B, we getz² =d²f (x²) + 2def (xy) + e²f (y²) = 0. We deduce thatf (xy) = 0, hencefis not injective, as desired.

On Problem 2013-1/A.We would like to thank Jan Stevens for pointing out that Problem 2013-1/A in fact originated from his work (see [2]), and that there is a connection with Coxeter’s frieze patterns (see [3, 4]).

References

1. B. Kostant and N. Wallach, Gelfand–Zeitlin theory from the perspective of clas- sical mechanics, Studies in Lie theory, Progress in Mathematics 243, 319–364, http://arxiv.org/abs/math/0408342v2.

2. J. Stevens, On the versal deformation of cyclic quotient singularities, Singularity theory and its applications, Part I, Lecture Notes in Math. 1462, Springer, Berlin, 1988/1989, 302–319.

3. J. H. Conway and H. S. M. Coxeter, Triangulated polygons and frieze patterns, Math. Gaz. 57 (1973), 87–94 and 175–183.

4. H. S. M. Coxeter, Frieze patterns, Acta Arith. 18 (1971), 297–310.