Béla Brindza(1958–2003)
Béla Brindza was born in 1958 in Csongrád, where he completed bothprimary and secondary school. His gift for mathematics became manifest
in his high school years. He participated with great success in national
and international competitions; in particular, he was a successful problem
solver of the “Középiskolai Matematikai Lapok”. His achievement at the
National Secondary School Mathematics Competition ensured his automatic admission to József Attila University (JATE) in Szeged. After his
ﬁrst year in Szeged he continued his studies at Kossuth Lajos University(KLTE) in Debrecen, from where he graduated in mathematics in 1983.
He completed his PhD thesis [2] in that same year, while still a student.
After graduation, with Kálmán Győry as his supervisor he ﬁnished anddefended his candidate’s thesis [9]. For his thesis [36] he was awarded theacademic doctoral degree in 1999.

2

Béla Brindza

He started teaching in his university years as a demonstrator. From1983 he was an instructor at KLTE until 1987, when he won a three year
postdoctoral scholarship at Macquarie University in Sydney. In 1990 he became an associate professor at KLTE, and then from 1992 to 1996 he taught
at Kuwait University. After his return in 1996 he was granted a scholarshipof the Hungarian National Scientiﬁc Foundation; and from 1997 he was associate professor at Kossuth University again. In 2000 he was appointed
full professor. He was also holder of the Széchenyi Professorial Scholarshipfrom 1998 to 2001. He reviewed articles regularly for the Mathematical Reviews and for the Zentralblatt für Mathematik. From 1992 he was memberof our periodical’s editorial board.
His scientiﬁc achievement was rewarded by the Kató Rényi Prize of theJános Bolyai Mathematical Association in 1982, the Géza Grünwald Prizein 1985, and the Academic Prize of the Hungarian Academy of Sciencesin 1992. He published his results in 48 papers, exposing them at a numberof universities and international conferences in Cambridge, Durham, Ann
Arbor, Oberwolfach, Leiden, Sydney, Tokyo and Kyoto. A whole chapter is devoted to Béla Brindza’s results in Shorey and Tijdeman’s bookExponential Diophantine Equations, a classical work on this subject.
Béla Brindza had an all-round talent for mathematics. Besides his
research work, he took an active part in mathematical life also at the levelof personal contacts. In particular, he paid a great deal of attention togifted young people. He conducted “student-olympics” study groups on
mathematics for high school students, while also guiding the professional
development of younger colleagues (Lajos Hajdu, Kálmán Liptai, László
Szalay, Attila Bérczes, Ákos Pintér). He was an outstanding lecturer, liked
and respected by his students. His special courses were lively and very
popular as he scintillated with wit.
Béla Brindza was an erudite, cheerful, sociable person, warm-hearted
and helpful. He had a passion for music. He was still very young when
his health betrayed him. A grave illness struck him down and he is nolonger with us. With his untimely death Hungarian Mathematics lost an
outstanding scientist, and the University of Debrecen an eminent professor.

Béla Brindza

3

The scientific research work of Béla Brindza was almost entirely
devoted to diophantine number theory. He made signiﬁcant contributions
to the theory of diophantine equations. His attention was mainly focused onderiving bounds for the solutions and the number of solutions of importantclassical equations which play a central rôle in modern diophantine analysis
and have a great number of applications. In Brindza’s papers modern,powerful methods, including the theory of linear forms in logarithms, weresuccessfully combined with new, original ideas and techniques elaborated
by him. In what follows, we give a brief survey of his outstanding scientiﬁcachievements.
1) Perhaps the most important results of Brindza are related to superelliptic equations. Let f (X) be a polynomial with integer coeﬃcients,and let r1 , . . . , rn denote the multiplicities of its zeros. For given m ≥ 2,
the superelliptic equationf (x) = y m

in integers x, y(1)

has only ﬁnitely many solutions, unless {m/(m, r1 ), . . . , m/(m, rn )} is apermutation of one of the n-tuples {t, 1, . . . , 1}, t ≥ 1, and {2, 2, 1, . . . , 1}.
This theorem, due to LeVeque (1964), was ineﬀective. Baker (1969) was
the ﬁrst to derive bounds for the solutions of (1), at last in the particularcase when m ≥ 3 and f possesses at least 2 simple zeros, or m = 2 and fpossesses at least 3 simple zeros. As a graduate student, Brindza [3] madeeﬀective LeVeque’s theorem in full generality. In fact he proved this in the
more general situation when the ground ring is the ring of S-integers of
a number ﬁeld. In the last two decades Brindza’s result has often beenquoted, treated and applied in the literature.
Schinzel and Tijdeman (1976) showed that if in (1) m is also unknown,
then m can be eﬀectively bounded above in terms of f , provided that f has
at least 2 distinct zeros. In his joint paper [41] with Bérczes and Hajdu,
Brindza gave an explicit upper bound for m in terms of the degree andheight of f . Further, together with Evertse and Győry he proved in [22]that, for irreducible f , m can be estimated from above only in terms ofthe degree and discriminant of f . These exerted considerable inﬂuence onlatter investigations of Haristoy, Győry, Pink, Pintér and others.
To extend the applicability of his results concerning (1), he showed in4Béla Brindza[12] that ifkN(X)f (X) = f1k1 (X) + . . . + fN

with coprime non-constant polynomials f1 , . . . , fN ∈ Z[X] such that µ =mini ki > N (N − 1), then f (X) has at least µ/(N − 1) distinct zeros. Thisimplies that for such a polynomial f (X) with N ≥ 2, m must be boundedin (1).
2) Brindza’s interest concerning (1) was greately motivated by Schäﬀer’s equation
Sk (x) = y m

in integers x, y > 1,(2)

where k and m > 1 are integers and Sk (x) = 1k +2k +. . .+xk . Since Sk (X)can be written as (Bk+1 (X + 1)− Bk+1 (0))/(k + 1) where Bk+1 (X) denotes
the (k +1)th Bernoulli polynomial, equation (2) is in fact a special superelliptic equation. Schäﬀer (1956) characterized those pairs (k, m) for which(2) has inﬁnitely many solutions. This was extended by Győry, Tijdeman
and Voorhoeve (1979) to the more general equation when in (2) m is also
unknown and Sk (x) is replaced by Sk (x)+R(x), where R(x) is any polynomial with coeﬃcients in Z. The above-mentioned results were ineﬀective.
By means of his general eﬀective theorem on equation (1) Brindza [4] succeeded to make eﬀective these results. Moreover, he generalized them in aneﬀective form for even more general equation of the shape F (Sk (x)) = y m ,where F ∈ (Z[X])[Y ] satisﬁes some natural conditions.
He derived also upper bounds for the number of solutions of (2). In[19] he proved that for any given m ≥ 3 with m = 4, equation (2) has at
most e7k solutions. Further, when in (2) the exponent m is also unknown
he proved with Pintér [44] that apart from the case (k, m) = (3, 4), equation (2) possesses at most max{c, e3k } solutions, where c is an eﬀectively
computable absolute constant.
In his nice joint article [47] with Bilu, Kirschenhofer, Pintér andTichy it is showed that for given k, l ≥ 2 and for suﬃciently large integer x, Sk (x) is not a product of l consecutive integers. Further, if k = l thenSk (x) = Sl (y) holds, provided x, y are suﬃciently large. To prove these,
the authors had to completely characterize the decomposable Bernoulli
polynomials, which is a result of independent interest.

Béla Brindza

53) A famous theorem of Tijdeman (1976) states that the Catalan equation
xm − y n = 1

in integers x, y, m, n > 1(3)

has only ﬁnitely many solutions, and all of them can be, at least in principle, eﬀectively determined. In [8], Brindza, Győry and Tijdeman extended this result to the number ﬁeld case when x, y are integers in anarbitrary but ﬁxed number ﬁeld K. They showed that if m, n > 1, mn > 4
and x, y are not roots of unity, then m, n and the heights of x, y can be
estimated from above by an eﬀective constant depending only on K. In
contrast with polynomial diophantine equations, the proof was not merely
a straightforward generalization of the rational case. In the general situation new arguments were needed to prove that m and n are bounded andthat x or y has a bounded norm. Later, Brindza [13] further generalized
this result for the cases of S-integers x, y of K.
Following a diﬀerent approach, recently Mihailescu proved that the
only solution of (3) is 32 − 23 = 1. It would be interesting to obtain similar
results in the more general cases considered by Brindza.
4) Many number theoretical problems can be reduced to unit equationsof the form
au + bv = c in u, v ∈ UK ,(4)

where a, b, c are given non-zero elements in a number ﬁeld K, and UK
denotes the unit group of K. By a theorem of Evertse, Győry, Stewart
and Tijdeman (1988) (4) has at most 2 solutions for “almost all” triples
of a, b, c, up to a proportional factor. In his joint paper [18] with Győry,
Brindza considerably improved this result in the important special casewhen a, b, c, are rational numbers. Then one may assume that a, b, c are
coprime positive integers and c ≥ a, b. In [18] it is proved that for all but
ﬁnitely many triples a, b, c of coprime positive integers, (4) has at mostone solution u, v (up to conjugacy), and u, v must belong to Q or a real
quadratic subﬁeld of K. It is also showed in an eﬀective form that if c is
suﬃciently large and ab > 1 or [K : Q] is odd, then no solution exists.
This surprising result was the ﬁrst application of the simultaneous Baker’s
method to diophantine equations.

6Béla Brindza

Eﬀective versions of Dirichlet’s unit theorem play an important rôlein applications of unit equations. In [24] Brindza employed a simple argument from the geometry of numbers to obtain a set of generators ofsmall height for the non-torsion subgroup of the S-unit group in a number
ﬁeld. His ingenious proof has exerted an inﬂuence on latter investigationsof Hajdu, Bugeaud, Győry and others.
5) Several remarkable results were obtained by Brindza on the Thue
equationF (x, y) = m in x, y ∈ Z,
(5)
where F is an irreducible binary form of degree n ≥ 3 with integer coeﬃcients and m is a non-zero integer. By Thue’s theorem (1909) (5) has onlyﬁnitely many solutions. The ﬁrst explicit bound for max{|x|, |y|} was given
by Baker (1968). This was later improved by several people. A surprisingly
good upper bound was derived by Brindza in terms of the height H(F )of F . Namely, he derived in [22] with Evertse and Győry a bound of
the form c1 H(F )3/n |D(F ) · m|c2 , where D(F ) denotes the discriminant of
F and c1 , c2 are eﬀectively computable numbers depending only on the
splitting ﬁeld of F .
There exist uniform upper bounds for the number of solutions of (5)which are independent of the coeﬃcients of F . Bombieri and Schmidt(1987) established e.g. the bound cns+1 , where c is an absolute constantand s denotes the number of distinct prime factors of m. Brindza [32]proved that if m is large enough with respect to |D(F )| then the numberof solutions does not exceedn(6s + 6).
It is a remarkable fact that this bound is already best possible in termsof n. In his joint paper [42] with Pintér, van der Poorten and Waldschmidt he obtained the bound 2n2 (s + 1) + 13n for the number of thosesolutions x, y for which max{|x|, |y|} is larger than a relatively small andexplicitly given bound which depends only on F and m. In other words,
only a “few” solutions of (5) can be “large”.
6) In [42], the following new gap principle of Brindza was the main
tool. Let t ∈ N and λ, η0 , . . . , ηt−1 , µ, ψ0 , . . . , ψt−1 ∈ C∗ . Suppose that theBéla Brindza

7

equation
λη k + µψ k = 1,
k

k

t−1
t−1
, ψ k = ψ0k0 . . . ψt−1
, has at least t + 2 solutions
where η k = η0k0 . . . ηt−1tkj = (k0,j , . . . , kt−1,j ) ∈ Z , j = 1, 2, . . . , t + 2. Let

K=

max

{2, |ki,j+1 − ki,j |}.

0≤i≤t−1, 1≤j≤t+1

If |λη k1 | ≥ 6 and |η kj +1 /η kj | ≥ 9(t + 1)t/2 K t for j = 1, . . . , t + 1, then(t + 1)t/2 K t /4 ≥ |λη k1 |.
In case of Thue equations this ensures exponential gaps between the solutions, provided that the equation has suﬃciently many solutions. We notethat the earlier gap principles yielded only polynomial gaps.
In [40] Brindza applied his gap principle to the generalized Ramanujan–Nagell equationf (x) = bz

in x, z ∈ Z, z > 1,

(6)where b ∈ Z with |b| > 1 and f ∈ Z[X] is of degree n with at least two
distinct zeros, one of which is not rational. He showed that the number ofsolutions of (6) is at most 6n2 (ns + n + 3)(s + 1), subject to the condition
that |b| is suﬃciently large. Here s denotes the number of distinct prime
factors of b. It is likely that Brindza’s gap principle will lead to further
applications as well.
7) Brindza was also interested in diophantine equations over function ﬁelds and ﬁnitely generated domains. Joining the investigations ofSchmidt, Győry, Mason and others, he established several nice results inthis direction.
Let K be a function ﬁeld in one variable over an algebraically closed
ﬁeld k of characteristic 0, S a ﬁnite set of places of K, and f (X) a polynomial with coeﬃcients in K and with zero multiplicities r1 , . . . , rn , respectively. As an analogue of his remarkable result over number ﬁelds, Brindza[10] derived together with Mason an explicit upper bound for the heights
of S-integral solutions x, y of (1), subject again to the condition that{m/(m, r1 ), . . . , m/(m, rn )} is not a permutation of either {t, 1, . . . , 1} or8Béla Brindza{2, 2, 1, . . . , 1}. Further, with Pintér and Végső he gave in [28] an explicit upper bound also for m, under the condition that in (1) n ≥ 2,
x, y ∈ K and y ∈/ k.
As a common generalization of Thue equations and superelliptic equations Brindza considered in [11] the equation
F (x, y) = cz m

in relatively prime S-integers x, y, z,(7)where F is a binary form with coeﬃcients in K and c ∈ K ∗ . Genaralizing
and improving a theorem of W. Schmidt (1980), he obtained an explicit
bound for the heights of the solutions of (7) in the case that F has at least
three distinct simple linear factors over the splitting ﬁeld of F . Moreover,he gave also an explicit upper bound for m when z ∈/ k.
Lang (1962) extended several classical diophantine ﬁniteness results
to that general situation when the ground ring R is any given ﬁnitely
generated integral domain over Z. Lang’s results were, however, ineﬀective. Győry (1983) worked out a method to establish eﬀective ﬁnitenesstheorems for Thue equations and more generally for decomposable form
equations over R. Combining this method with his own results obtained
over number ﬁelds and function ﬁelds Brindza [17], [27] generalized his
eﬀective ﬁniteness theorems on equations (1) and (3) for the ﬁnitely generated case.
8) Finally, let us mention an interesting result of Brindza in algebraic
coding theory. Brindza’s attention was drawn to the following problem by
K. Buzási. Let K be a ﬁnite ﬁeld of characteristic 2, and KG the groupalgebra of G over K where G = (a) × (b) and ap = bp = 1 for an odd
prime p. Then KG = I0 ⊕ . . . ⊕ Ip+1 , where each In is a minimal ideal
generated by en , n = 0, 1, . . . , p + 1,
e0 =


g∈G

and

g,

p−1



ep+1 =
bi a + . . . + ap−1
i=0

p−1

 n−1 i abb + . . . + bp−1 ,en =n = 1, . . . , p.

i=0Brindza [6] as a graduate student disproved an old conjecture of S. D. Berman (1967) by proving that if p ≡ −1 (mod 4) and if I ⊂ KG is an idealBéla Brindza9

which is the sum of three minimal ideals diﬀerent from I0 and Ip+1 , thenthe code distance d(I) is equal to 2(p − 1).

List of publications of B. Brindza[1] B. Brindza, The functions rk (n) and problems of Paul Erdős, Mat. Lapok 30(1978–82), 127–133 (in Hungarian).
[2] B. Brindza, Investigations in number theory and algebraic coding theory, University doctoral dissertation, Debrecen, 1983 (in Hungarian).
[3] B. Brindza, On S-integral solutions of the equation y m = f (x), Acta Math. Hungar. 44 (1984), 133–139.
[4] B. Brindza, On some generalizations of the diophantine equation 1k + 2k + . . .
· · · + xk = y z , Acta Arith. 44 (1984), 99–107.
[5] B. Brindza, On a diophantine equation connected with the Fermat equation, Acta
Arith. 44 (1984), 357–363.
[6] B. Brindza, On the minimal distance of (P, P ) codes, Information Theory, Moscow
21 (1985), 10–19 (in Russian).
[7] B. Brindza, K. Győry and R. Tijdeman, The Fermat equation with polynomial
values as base variables, Invent. Math. 80 (1985), 139–151.
[8] B. Brindza, K. Győry and R. Tijdeman, The Catalan equation over algebraic
number ﬁelds, J. Reine Angew. Math. 367 (1986), 90–102.
[9] B. Brindza, Hyperelliptic diphantine equations and their applications, Candidate’s
dissertation, Debrecen, 1986 (in Hungarian).
[10] B. Brindza and R. C. Mason, LeVeque’s superelliptic equation over function
ﬁelds, Acta Arith. 47 (1986), 167–173.
[11] B. Brindza, On the equation F (x, y) = z m over function ﬁelds, Acta Math. Hungar.
49 (1987), 267–275.
[12] B. Brindza, Zeros of polynomials and exponential diophantine equations, Compositio Math. 61 (1987), 137–157.
[13] B. Brindza, On S-integral solutions of the Catalan equation, Acta Arith. 47 (1987),
397–412.
[14] B. Brindza and I. Gaál, Inhomogeneous norm form equations in two dominating
variables over function ﬁelds, Acta Math. Hungar. 50 (1987), 147–153.
[15] B. Brindza, Some new applications of a theorem of Mason, in: New Advances inTranscendence Theory, Cambridge Univ. Press, Cambridge, 1988, 82–89.
[16] B. Brindza and M. Waldschmidt, Estimates for linear forms in logarithms andtheir applications, J. Math. Madras Univ. 51 (1988), 147–160.
[17] B. Brindza, On the equation f (x) = y m over ﬁnitely generated domains, Acta
Math. Hungar. 53 (1989), 377–383.
[18] B. Brindza and K. Győry, On unit equations with rational coeﬃcients, Acta
Arith. 53 (1989), 367–388.

10Béla Brindza[19] B. Brindza, Power values of the sum 1k + 2k + . . . + xk , in: Number Theory (Budapest, 1987), Coll. Math. Soc. János Bolyai, Vol. 51, North-Holland Publ. Comp.,
1990, 595–603.
[20] B. Brindza, Thue equations and multiplicative independence, in: Cryptography
and Number Theory, (J. Loxton, ed.), Cambridge University Press, Cambridge,
1990, 213–220.
[21] B. Brindza and P. Erdős, On some diophantine problems involving powers andfactorials, J. Austral. Math. Soc. 51 (1991), 1–7.
[22] B. Brindza, J. H. Evertse and K. Győry, Bounds for the solutions of somediophantine equations in terms of their discriminant, J. Austral. Math. Soc. 51(1991), 8–26.
´ Pintér, On the power values of Stirling numbers, Acta Arith.
[23] B. Brindza and A.
60 (1991), 169–175.
[24] B. Brindza, On the generators of S-unit groups in algebraic number ﬁelds, Bull.
Austral. Math. Soc. 43 (1991), 325–329.
[25] B. Brindza, On a Special Superelliptic Equation, Publ. Math. Debrecen 39 (1991),
159–162.
[26] B. Brindza, On the number of solutions of Thue’s equation, in: Sets, Graphs andNumbers (Budapest, 1991), Coll. Math. Soc. J. Bolyai, Vol. 60, North-Holland,
1992, 127–133.
[27] B. Brindza, The Catalan equation over ﬁnitely generated integral domains, Publ.
Math. Debrecen 42 (1993), 193–198.
´ Pintér and J. Végső, The Schinzel–Tijdeman theorem over func[28] B. Brindza, A.
tion ﬁelds, C. R. Math. Rep. Acad. Sci. Canada 16 (1994), 53–58.
´ Pintér, On equal values of binary forms over ﬁnitely generated[29] B. Brindza and A.
ﬁelds, Publ. Math. Debrecen 46 (1995), 339–347.
´ Pintér, On positive integer solutions of[30] B. Brindza, Al-Zaid Hassan and A.
the equation xy + yz + xz = n, Canad. Math. Bull. 39 (1996), 199–202.
´ Pintér, On equal values of power sums, Acta Arith. 77 (1996),[31] B. Brindza and A.
97–101.
[32] B. Brindza, On large values of binary forms, Rocky Mount. Math. J. 26 (1996),
839–845.
´ Pintér and J. Végső, On polynomial values of the discriminant[33] B. Brindza, A.
of characteristic polynomials, J. Number Theory 61 (1996), 292–300.
[34] B. Brindza, A. Borbély, V. Bovdi and T. Krausz, Covering theorems for Artinian rings, Publ. Math. Debrecen 51 (1997), 323–329.
´ Pintér, On the irreducibility of some polynomials in two[35] B. Brindza and A.
variables, Acta Arith. 82 (1997), 303–307.
[36] B. Brindza, Estimates for the number of solutions of diophantine equations, 1997,
Academic doctoral dissertation.
´ Pintér, On the Kronecker–Győry specialization, in: Proc. of[37] B. Brindza and A.
Number Theory Conf. (Eger, 1996), 1998, 107–113.

Béla Brindza

11[38] B. Brindza, K. Liptai and L. Szalay, On products of the terms of linear recurrences, in: Proc. of Number Theory Conf., (Eger, 1996), 1998, 101–106.
´ Pintér and S. Turjányi, On equal values of polygonal and[39] B. Brindza, A.
pyramidal numbers, Indag. Math. 9 (1998), 183–185.
[40] B. Brindza, On a generalization of the Ramanujan–Nagell equation, Publ. Math.
Debrecen 53 (1998), 225–233.
[41] B. Brindza, A. Bérczes and L. Hajdu, On power values of polynomials, Publ.
Math. Debrecen 53 (1998), 375–381.
´ Pintér, Alf van der Poorten and M. Waldschmidt, On the[42] B. Brindza, A.
distribution of solutions of Thue’s equations, in: Proc. of Number Theory Conf.
(Zakopane, 1997), 1999, 35–46.
[43] B. Brindza, L. Hajdu and I. Ruzsa, On the equation
x(x + d) . . . (x + (k − 1)d) = by 2 , Glasgow Math. J. 42 (2000), 255–261.
´ Pintér, On the number of solutions of the equation 1k +2k +. . .
[44] B. Brindza and A.
k· · · + (x − 1) = y z , Publ. Math. Debrecen 56 (2000), 271–277.
´ Pintér and W. M. Schmidt, Multiplicities of binary recurrences,[45] B. Brindza, A.
Canadian Math. Bull. 44 (2001), 19–21.
[46] B. Brindza, Remarks on the number of non-zero coeﬃcients of polynomials, Acta
Math. Acad. Paedagog. Nyházi. 17 (2001), 77–79.
´ Pintér and R. Tichy, Dio[47] B. Brindza, Yu. Bilu, P. Kirschenhofer, A.
phantine equations and Bernoulli polynomials, With an appendix by A. Schinzel,
Compositio Math. 131 (2002), 173–188.
[48] B. Brindza and Gy. Maksa, A note on non-negative information functions, Acta
Acad. Paedagog. Agriensis 30 (2003), 31–37.

K. Győry, A. Pethő and Á. Pintér