Magic Squares of Squares by Philippe Michaud-Rodgers MA4K9 Dissertation Submitted to The University of Warwick Mathematics Institute April, 2019 Table of contents 1 Introduction 1 2 Hilbert Polynomials and Dimension 3 2.1 Defining the Dimension of a Variety . .4 2.2 The Complement of a Monomial Ideal . .7 2.3 Dimension of a Projective Variety and an Explicit Calculation . .9 3 Primary Decomposition and Singular Points 11 3.1 Primary Decomposition . 11 3.2 Singular Points . 12 4 Grassmannians 17 4.1 General Theory . 17 4.2 Local Coordinates on Gr(2; n +1) ....................... 20 5 Lines, Curves, and Variations on the Magic Square 23 5.1 Lines on Varieties and Examples . 23 5.2 The Magic Hourglass and Curves on Varieties . 29 5.3 Near Magic Squares of Squares . 35 6 Conclusion 37 A MATLAB Code 39 ii 1 Introduction The study of magic squares dates back to ancient China and has a rich history. According to Chinese legend [6, p. 38], the mythical emperor Yu found a turtle along the banks of the river Luo with an unnatural diagram on the back of its shell (Figure 1) which he named the Lo Shu. Figure 1: The ancient image of the Chinese Lo Shu [15]. His discovery was that of a magic square, a 3 × 3 grid with the numbers one to nine placed in the grid such that the sum of the numbers in each row, column, and main diagonal is equal. Here is the Lo Shu magic square in modern notation. 4 9 2 3 5 7 8 1 6 This magic square was known to Chinese mathematicians as far back as 190 BCE, but was important more for its philosophical and mystical properties than its mathematical ones [6]. Magic squares were also studied extensively in the Middle East and India, mainly from around 600-1400 CE [7, 19], before gaining popularity with European mathematicians around the 16th century [5, p. 170]. Although the traditional magic square consists of placing the numbers 1 to 9 in a 3 × 3 grid, by considering different grid sizes (n × n for n ≥ 2), by allowing any numbers to be placed in the grid (not just from 1 to n2), and by imposing certain additional constraints, the theory of magic squares and the questions one may ask become much deeper. In particular, a fairly natural question is: does there exist a magic square of squares with nine distinct entries? That is, a magic square whose entries are distinct square numbers. There are of course some trivial examples by allowing repeated entries, and so we will often tacitly assume that we are looking for solutions with distinct entries. It would seem that Euler was the first mathematician to study this question seriously, and managed to construct a 4 × 4 magic square of squares, as well as a 3 × 3 semi-magic square of squares (a grid in which the row and column sums are all equal, but the diagonals may not add up to this same number). A century later, in 1876, Edouard Lucas studied the problem 1 in the 3 × 3 case, and although he made some progress, he could not reach an answer [2, p. 55]. The problem then gained a renewed popularity in the 20th century after Martin Labar asked, in 1984, for either a proof or disproof of the existence of a 3 × 3 magic square of squares, and subsequently when Martin Gardner republished the same question in 1996 [2, p. 52]. Two years later, Gardner wrote: So far no one has come forward with a `square of squares' { but no one has proved its impossibility either. If it exists, its numbers would be huge, perhaps beyond the reach of today's fastest computers. [11, p. 74] Indeed, it has been shown [4] that if such a magic square exists, then its central entry must be greater than 25 × 1024, and computer searches lead us to believe that no such magic square can exist. A full historical overview of the magic square of squares problem can be found in [2]. Conjecture 1.1. There does not exist a 3 × 3 magic square consisting of nine distinct square numbers. This conjecture has been studied in quite some depth, although almost solely from the point of view of elementary number theory and using traditional methods of constructing magic squares such as those employed by Lucas. Here, we study the problem by linking the set of magic squares of squares with an algebraic variety, and investigating its properties. We view a magic square as a matrix in the variables a; b; c; d; e; f; g; h; i in a field k: 0 1 a b c B C Bd e fC. @ A g h i We do not specify the field k in which we are working, although we will assume that k is some extension of Q, usually C. We are interested in solutions (or their non-existence) over the rationals, as then a rescaling will give us integer entries. For this matrix to be a magic square of squares once we square each entry, we require that a2 + b2 + c2 = d2 + e2 + f 2 = g2 + h2 + i2 = a2 + d2 + g2 = b2 + e2 + h2 = c2 + f 2 + i2 = a2 + e2 + i2 = c2 + e2 + g2. Equivalently, we can express these defining equations as: a2 + b2 + c2 − d2 − e2 − f 2 = 0 a2 + b2 + c2 − g2 − h2 − i2 = 0 a2 + b2 + c2 − a2 − d2 − g2 = 0 a2 + b2 + c2 − b2 − e2 − h2 = 0 (1) a2 + b2 + c2 − c2 − f 2 − i2 = 0 a2 + b2 + c2 − a2 − e2 − i2 = 0 a2 + b2 + c2 − c2 − e2 − g2 = 0: 2 Since a rescaling of a magic square is still a magic square, and we would like to study the solution sets of the above equations using techniques from algebraic geometry, we will 8 work in projective space; in particular in P as we have nine variables. Since each of the polynomials appearing on the left-hand side of the equations (1) is homogeneous, we can define the following. Definition 1.1. We define the magic square variety, which we denote X, to be the pro- 8 jective variety in P given as: n 8 o X := (a : b : c : d : e : f : g : h : i) 2 P j (1) holds : (2) Given a point on this variety, we obtain a magic square of squares by squaring each entry. Our goal is to study the geometric properties of this variety. In Section 2, we introduce Hilbert polynomials in order to calculate the dimension of X, as well as its degree. In Section 3, we briefly introduce primary decomposition and then look at singular points, giving a complete classification of the singular points of the magic square variety. In Section 4, we introduce the theory of Grassmannians using the exterior algebra, and then in Section 5, we use this theory to consider lines and curves on X and variations on this variety. We also find solutions over number fields of low degree using these methods. Throughout, we use the computer algebra system Macaulay2 [12] to carry out the calculations that are too difficult or too long to do by hand. 2 Hilbert Polynomials and Dimension Our first goal is to understand the dimension of our magic square variety, X. The dimension of an algebraic variety can be defined in several equivalent ways, all of which are rather complicated! In this section we introduce the notion of Hilbert polynomials as a means of defining dimension. Intuitively, the dimension of a variety is what it looks like `close up'. Take for example a 3 sphere in R ; then by zooming in close enough to the surface of the sphere, it looks more or less like a plane, and so we would say that the sphere is 2-dimensional. If our variety were reducible, say the union of a sphere and a line, then we would say it has dimension equal to the maximum of the dimensions of its irreducible components, in this case 2. Despite this very intuitive point of view, in order to define the dimension of a variety through the language of commutative algebra, we go via a different route. Throughout this section we follow the book Ideals, Varieties, and Algorithms by David Cox, John Little, and Donal O'Shea [8, pp. 439{468]. We concentrate on understanding the ideas and presenting examples rather than presenting the proofs which can be found in the book. We focus on the case of affine varieties, although the results that we state 3 for projective varieties in Subsection 2.3 are very similar. In this section we work over an algebraically closed field, k. 2.1 Defining the Dimension of a Variety Let V ⊆ kn be an affine variety. One way of understanding the geometry of V is by investigating the functions on V . We define the coordinate ring of V , which we denote k[V ], as k[x ; : : : ; x ] k[V ] := 1 n : I(V ) This is also a vector space over k. Here, I(V ) is the set of polynomials vanishing on V (which forms an ideal). We will also write V (J) for the common zero locus of a set of polynomials J ⊆ k[x1; : : : ; xn]. Rather than think of k[V ] as a quotient ring, we view elements of k[V ] as polynomials restricted to V , since if two polynomials take the same values on V , then their difference lies in I(V ). A first guess of how to define the dimension of V may be to set it as the dimension of k[V ], and this is along the right lines, but k[V ] will usually be infinite-dimensional and so this is not quite right.