Shalom Eliahou

Infinite families of links with trivial Jones polynomial

Abstract For each k⩾2, we exhibit infinite families of prime k-component links with Jones polynomial equal to that of the k-component unlink.

On Golay polynomial pairs

In this article, we study Golay polynomial pairs, a notion arising in various fields such as optics, engineering, combinatorics, and Fourier analysis.

Optimally small sumsets in finite abelian groups

Abstract Let G be a finite abelian group of order g. We determine, for all 1⩽r,s⩽g, the minimal size μG(r,s)=min|A+B| of sumsets A+B, where A and B range over all subsets of G of cardinality r and s, respectively. We do so by explicit construction. Our formula for μG(r,s) shows that this function only depends on the cardinality of G, not on its specific group structure. Earlier results on μG are recalled in the Introduction.

On systems of binomials in the ideal of a toric variety

Let r be a toric set in the affine space A n k . Given a set of binomials g 1 ,…, g r in the toric ideal P of Γ, we give a criterion for deciding the equality rad(g 1 ,…,g r ) = P. This criterion extends to arbitrary dimension, and to arbitrary fields, an earlier result which concerned only monomial curves over an algebraically closed field of characteristic zero.

Signed Diagonal Flips and the Four Color Theorem

We introduce a signed version of the diagonal flip operation. We then formulate the conjecture that any two triangulations of a given polygon may be transformed into one another by a signable sequence of diagonal flips. Finally, we show that this conjecture, if true, would imply the four color theorem.

On a Problem of Steinhaus Concerning Binary Sequences

A finite ±1 sequence X yields a binary triangle ΔX whose first row is X, and whose (k + 1)th row is the sequence of pairwise products of consecutive entries of its kth row, for all k ≥ 1. We say that X is balanced if its derived triangle ΔX contains as many +1s as −1s. In 1963, Steinhaus asked whether there exist balanced binary sequences of every length n ≡ 0 or 3 mod 4. While this problem has been solved in the affirmative by Harborth in 1972, we present here a different solution. We do so by constructing strongly balanced binary sequences, i.e., binary sequences of length n all of whose initial segments of length n – 4t are balanced, for 0 ≤ t ≤ n/4. Our strongly balanced sequencesdo occur in every length n ≡ 0 or 3 mod 4. Moreover, we provide a complete classification of sufficiently long strongly balanced binary sequences.

The 3 x +1 problem: new lower bounds on nontrivial cycle lengths

Abstract Let T: N → N be the function defined by T(n)= n 2 if n is even, T(n)= (3n+1) 2 if n is odd. We show, among other things, that any nontrivial cyclic orbit under iteration of T must contain at least 17087915 elements.

Some extensions of the Cauchy-Davenport theorem

Abstract The Cauchy-Davenport theorem states that, if p is prime and A, B are nonempty subsets of cardinality r, s in Z / p Z , the cardinality of the sumset A + B = { a + b | a ∈ A , b ∈ B } is bounded below by min ( r + s − 1 , p ) ; moreover, this lower bound is sharp. Natural extensions of this result consist in determining, for each group G and positive integers r , s ⩽ | G | , the analogous sharp lower bound, namely the function μ G ( r , s ) = min { | A + B | | A , B ⊂ G , | A | = r , | B | = s } . Important progress on this topic has been achieved in recent years, leading to the determination of μ G for all abelian groups G. In this note we survey the history of earlier results and the current knowledge on this function.

Sumsets in dihedral groups

Let Dn be the dihedral group of order 2n. For all integers r,s such that 1 ≤ r,s ≤ 2n, we give an explicit upper bound for the minimal size µDn (r, s) = min |A ˙ B| of sumsets (product sets) A ˙ B, where A and B range over all subsets of Dn of cardinality r and s respectively. It is shown by construction that µDn (r, s) is bounded above by the known value of µG(r, s), where G is any abelian group of order 2n. We conjecture that this upper bound is sharp, and prove that it really is if n is a prime power.

Symbolic powers of monomial curves

The answer is positive for ideal-theoretic local complete intersection (e.g., smooth) curves [S], [MK], and for arbitrary curves if char(k) is non-zero [C-N]. Problem 1 remains open for “most” singular curves over (say) C. In fact, we look at an algebraic problem related to Problem 1. Let P be a prime ideal in a commutative noetherian ring A. Denote by S(P) the direct sum ora0 P”) of the symbolic powers of P (see Section 0 for definitions), which is canonically a graded A-algebra. It is called the symbolic Rees algebra of P (or of X, if X is the variety defined by P in Spec A).