Quadratic forms
Significant results in mathematics that are also not highly technical come out too seldomly. So it was nice to see the report by Ivars Peterson in the March 11, 2006 issue of Science News concerning striking recent discoveries by the young mathematician Manjul Bhargava that have to do with long-standing questions in the arithmetic theory of quadratic forms.
The article can be found here or here.
As good as the article is, some readers may be curious about a few relatively simple details which weren't explained.
To begin with, let's have a general definition of quadratic form, that is, a polynomial in 1 or more variables in which every term has degree exactly two. The situation is pretty trivial with only one variable, so normally only cases with two or more variables are considered.
With two variables, x and y, the most general quadratic form would be f(x,y) = ax2 + bxy + cy2, where the coefficients a, b, and c are numbers rather than variables. To say that all terms have degree two means that the sum of exponents of the variables in one term is two. Another way to express this condition is to say that the polynomial is homogeneous of degree two.
There could be any positive integer number n of variables, in which case the form is said to be n-ary. For small n, the terminology binary, ternary, or quaternary (n = 2, 3, or 4) is used.
An n-ary quadratic form can be written symbolically, using the summation operator Σ, as
The numbers qij are the coefficients of the form. The condition i≤j on the indices in the summation is so that terms like xixj and xjxi are included only once if i ≠ j. This sensible convention actually has interesting consequences, as we'll see in a moment.
The coefficients, in general, could be any sort of numbers. But for purposes of number theory (which is what concerns us here), the coefficients are assumed to be positive integers.
A natural question one may ask in this case is whether a given positive integer n can be represented by a quadratic form, that is, whether the quadratic form takes on the value n for some integer values of the variables. For example, in the simple case of two variables, this amounts to solving the Diophantine equation n = ax2 + bxy + cy2, where a, b, and c are integers.
It turns out that in order to study the algebra of quadratic forms it is helpful to represent them not as polynomials but as matrices A = (qij) that contain the coefficients of the form. (I won't explain matrices here, so if you haven't encountered the idea, it's best to skip ahead.) Then if x = (x1, ... , xn) is a row vector whose entries consist of the variable symbols, and xT is its transpose (a column vector), another way to write the quadratic form is as the matrix product xTAx = f(x1, ... , xn). Unfortunately, there's a kink in this scheme. The matrix will contain coefficients qij for all values of i and j between 1 and n. The matrix needs to be symmetric, so that qij = qji. But then products like xixj get counted twice if i ≠ j, so for the off-diagonal entries of the matrix A it's necessary to use qij/2 when i ≠ j. And that's all well and good, unless we're interested in forms with integer coefficients and we need the matrix entries to be integers as well.
To illustrate the problem, consider the form x2+2xy+y2. The matrix that corresponds to this is the 2×2 square matrix whose entries are all 1. However, for the form x2+xy+y2 the corresponding matrix has 1/2 as the off-diagonal entries. For some purposes that's OK, but for number theory we don't want non-integral matrix elements. So one says that the first form in this example, which has a corresponding matrix with integral entries, is matrix-defined, while the second form is not. (Mathematicians are very picky.)
With these preliminaries out of the way, let's look at what Bhargava (and others) have proved. In 1770 Joseph-Louis Lagrange, one of the earliest number theorists, proved that every positive integer is the sum of at most four non-zero squares. Another way to say this is that the quaternary quadratic form f(x,y,z,w) = x2 + y2 + z2 + w2 takes on every possible positive integer value for some value of the variables x, y, z, w. Or, using a term mentioned before, the form represents every positive integer. A quadratic form with this property is said to be universal.
Mathematicians are always looking for generalizations, so the next question one might ask is whether every quaternary quadratic form is universal. That's obviously not the case, since a form like 2x2 + 2y2 + 2z2 + 2w2 can represent only even numbers. So the next, but much harder, question is whether there are relatively simple conditions that can identify forms that are universal. Indeed, are there any other forms that are universal, besides Lagrange's? In 1916, the extraordinary Indian mathematician Srinivasa Ramanujan found 53 more universal quaternary quadratic forms with integer coefficients.
Much later, in 1993, John H. Conway and William Schneeberger proved that there's a prety simple condition that guarantees a quadratic form is universal, provided it's also matrix-defined, as explained above. The condition is that in order to be universal and represent every positive integer, it's sufficient for the form to represent just the nine integers 1, 2, 3, 5, 6, 7, 10, 14, and 15. This theorem became known as the "15 theorem". (Conway is an extremely versatile and interesting mathematician himself. Among his other accomplishments are invention of the "Game of Life", the discovery of three "sporadic" finite simple groups, and the discovery (with Simon Kochen) of a "free-will" theorem related to the EPR paradox of quantum mechanics.)
Lagrange's form as well as all 53 discovered by Ramanujan are of the matrix-defined type. So Conway's 15 theorem quickly allows for a proof that all are universal. The next thing one might ask is whether there are only a finite number of universal matrix-defined forms, and if so, exactly how many there are. This is where Bhargava comes in. In the late 1990s, working with Conway as a graduate student at Princeton, Bhargava found a new, simpler proof of the 15 theorem (whose original proof was long and complicated). Using his new ideas, he then proved that there ara exactly 204 different universal matrix-defined quadratic forms. He also proved related results, such as that a matrix-defined form represents all odd numbers if it can represent just 1, 3, 5, 7, 11, 15, and 33. Similarly, a matrix-defined form can represent all prime numbers if it represents just the primes up to 73.
The next step is to try to relax the condition which requires the form to be matrix-defined in order for simple criteria to be applicable. One would like a criterion that can be used for any quadratic form which takes only integer values, even it's not matrix-defined. Just recently Bhargava, working with Jonathan P. Hanke of Duke University, has come up with an answer.
What they found is that there is a set of 29 integers less than or equal to 290 such that a quaternary integer-valued quadratic form is universal if it represents each of those 29 integers. (Conway and Schneeberger had earlier conjectured that testing all numbers from 1 to 290 would be sufficient.) Then, using a computer to check all the cases that could occur, Bhargava and Hanke found that there are exactly 6436 universal, integer-valued, quaternary quadratic forms.
Tags: mathematics, number theory, quadratic form
The article can be found here or here.
As good as the article is, some readers may be curious about a few relatively simple details which weren't explained.
To begin with, let's have a general definition of quadratic form, that is, a polynomial in 1 or more variables in which every term has degree exactly two. The situation is pretty trivial with only one variable, so normally only cases with two or more variables are considered.
With two variables, x and y, the most general quadratic form would be f(x,y) = ax2 + bxy + cy2, where the coefficients a, b, and c are numbers rather than variables. To say that all terms have degree two means that the sum of exponents of the variables in one term is two. Another way to express this condition is to say that the polynomial is homogeneous of degree two.
There could be any positive integer number n of variables, in which case the form is said to be n-ary. For small n, the terminology binary, ternary, or quaternary (n = 2, 3, or 4) is used.
An n-ary quadratic form can be written symbolically, using the summation operator Σ, as
f(x1, ... , xn) = Σ1≤i≤j≤n qijxixj
The numbers qij are the coefficients of the form. The condition i≤j on the indices in the summation is so that terms like xixj and xjxi are included only once if i ≠ j. This sensible convention actually has interesting consequences, as we'll see in a moment.
The coefficients, in general, could be any sort of numbers. But for purposes of number theory (which is what concerns us here), the coefficients are assumed to be positive integers.
A natural question one may ask in this case is whether a given positive integer n can be represented by a quadratic form, that is, whether the quadratic form takes on the value n for some integer values of the variables. For example, in the simple case of two variables, this amounts to solving the Diophantine equation n = ax2 + bxy + cy2, where a, b, and c are integers.
It turns out that in order to study the algebra of quadratic forms it is helpful to represent them not as polynomials but as matrices A = (qij) that contain the coefficients of the form. (I won't explain matrices here, so if you haven't encountered the idea, it's best to skip ahead.) Then if x = (x1, ... , xn) is a row vector whose entries consist of the variable symbols, and xT is its transpose (a column vector), another way to write the quadratic form is as the matrix product xTAx = f(x1, ... , xn). Unfortunately, there's a kink in this scheme. The matrix will contain coefficients qij for all values of i and j between 1 and n. The matrix needs to be symmetric, so that qij = qji. But then products like xixj get counted twice if i ≠ j, so for the off-diagonal entries of the matrix A it's necessary to use qij/2 when i ≠ j. And that's all well and good, unless we're interested in forms with integer coefficients and we need the matrix entries to be integers as well.
To illustrate the problem, consider the form x2+2xy+y2. The matrix that corresponds to this is the 2×2 square matrix whose entries are all 1. However, for the form x2+xy+y2 the corresponding matrix has 1/2 as the off-diagonal entries. For some purposes that's OK, but for number theory we don't want non-integral matrix elements. So one says that the first form in this example, which has a corresponding matrix with integral entries, is matrix-defined, while the second form is not. (Mathematicians are very picky.)
With these preliminaries out of the way, let's look at what Bhargava (and others) have proved. In 1770 Joseph-Louis Lagrange, one of the earliest number theorists, proved that every positive integer is the sum of at most four non-zero squares. Another way to say this is that the quaternary quadratic form f(x,y,z,w) = x2 + y2 + z2 + w2 takes on every possible positive integer value for some value of the variables x, y, z, w. Or, using a term mentioned before, the form represents every positive integer. A quadratic form with this property is said to be universal.
Mathematicians are always looking for generalizations, so the next question one might ask is whether every quaternary quadratic form is universal. That's obviously not the case, since a form like 2x2 + 2y2 + 2z2 + 2w2 can represent only even numbers. So the next, but much harder, question is whether there are relatively simple conditions that can identify forms that are universal. Indeed, are there any other forms that are universal, besides Lagrange's? In 1916, the extraordinary Indian mathematician Srinivasa Ramanujan found 53 more universal quaternary quadratic forms with integer coefficients.
Much later, in 1993, John H. Conway and William Schneeberger proved that there's a prety simple condition that guarantees a quadratic form is universal, provided it's also matrix-defined, as explained above. The condition is that in order to be universal and represent every positive integer, it's sufficient for the form to represent just the nine integers 1, 2, 3, 5, 6, 7, 10, 14, and 15. This theorem became known as the "15 theorem". (Conway is an extremely versatile and interesting mathematician himself. Among his other accomplishments are invention of the "Game of Life", the discovery of three "sporadic" finite simple groups, and the discovery (with Simon Kochen) of a "free-will" theorem related to the EPR paradox of quantum mechanics.)
Lagrange's form as well as all 53 discovered by Ramanujan are of the matrix-defined type. So Conway's 15 theorem quickly allows for a proof that all are universal. The next thing one might ask is whether there are only a finite number of universal matrix-defined forms, and if so, exactly how many there are. This is where Bhargava comes in. In the late 1990s, working with Conway as a graduate student at Princeton, Bhargava found a new, simpler proof of the 15 theorem (whose original proof was long and complicated). Using his new ideas, he then proved that there ara exactly 204 different universal matrix-defined quadratic forms. He also proved related results, such as that a matrix-defined form represents all odd numbers if it can represent just 1, 3, 5, 7, 11, 15, and 33. Similarly, a matrix-defined form can represent all prime numbers if it represents just the primes up to 73.
The next step is to try to relax the condition which requires the form to be matrix-defined in order for simple criteria to be applicable. One would like a criterion that can be used for any quadratic form which takes only integer values, even it's not matrix-defined. Just recently Bhargava, working with Jonathan P. Hanke of Duke University, has come up with an answer.
What they found is that there is a set of 29 integers less than or equal to 290 such that a quaternary integer-valued quadratic form is universal if it represents each of those 29 integers. (Conway and Schneeberger had earlier conjectured that testing all numbers from 1 to 290 would be sufficient.) Then, using a computer to check all the cases that could occur, Bhargava and Hanke found that there are exactly 6436 universal, integer-valued, quaternary quadratic forms.
Tags: mathematics, number theory, quadratic form
Labels: mathematics
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1 Comments:
Since I read the Science News article yesterday, I have been feeling good about life in general.
Funny how invigorating it is to learn of certain breakthroughs, particularly at an elementary level. I haven't read Bhargava's proofs, but this whole thing sounds as if it transpired without use of the monstrous machinery of modern quadratic-form theory. Elegant elementary approaches are still out there, we're just trained not to see them -- plus not so brilliant as that fellow obviously is.
Great stuff.
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