Dr. Matthew P. Skerritt

RMIT University

65th Annual Meeting of the Australian Mathematics Society.

Newcastle, NSW, December 7–10th 2021

Newcastle, NSW, December 7–10th 2021

For convenience, we will usually think of \( x \in \) \( \RR^n \) and \( a \in \) \( \ZZ^n \) and say that \( a \) is an integer realtion of \( x \).

For convenience, we will usually think of \( x \in \) \( \CC^n \) and \( a \in \) \( \ZZ[i]^n \) and say that \( a \) is an integer realtion of \( x \).

When talking about integer relations in general, we use the following notation so as to encapsulate all cases.

We will use \( \FF \) to denote the field of numbers which we are relating and we will use \( \mathcal{O} \) to denote the ring of integers from which the relation elements are from.

The previous definitions we call the *classical* cases. That is, when \( \mathcal{O} = \ZZ \), \( \FF = \RR\), and when \( \mathcal{O} = \ZZ[i] \), \( \FF = \CC \).

An information theoretic argument shows that in order to numerically compute an integer relation on \( n \) numbers, where the relation consists of integers with no more than \( d \) digits, then at least \( nd \) decimal digits of precision need to be used.

In the case of complex numbers and Gaussian integers, the number of digits is the number of digits in the real part, or the imaginary part, whichever is larger.

The classical cases can be computed with

- LLL (indirectly)
- PSLQ (directly)

The LLL algorithm was introduced by Lenstra, Lenstra Jr, and Lovasz in 1982.

It is not an algorithm for computing integer relations directly. Instead it finds short lattice vectors

It can, however, be used to compute integer relations

Let \( B = \left\{ b_1, \dotsc, b_n\right\} \subset \FF^n \) be a linearly independent set of vectors. The set \[ L = \left\{ \lambda_1 b_1, \dotsc, \lambda_n b_n : \lambda_1 \in \mathcal{O} \right\} \] is the *lattice spanned by \(B\)*. We call \( B \) the *basis* of the lattice \( L \)

We will sometimes refer to an \( \mathcal{O} \)-lattice if we need to specify the integer ring explicitly.

Given a basis \( B \) for a lattice \( L \), the LLL algorithm computes a basis \( B^\prime = (b^\prime_1,\dotsc,b^\prime_n) \) such that \[ \begin{gather} \lvert \mu_{i,j} \rvert \le \frac{1}{2} \text{ for } 1 \le j < i \le n \\ \lvert\lvert b^*_i + \mu_{i,j} b^*_{i-1} \rvert\rvert^2 \ge \frac{3}{4} \lvert\lvert b^*_{i-1} \rvert\rvert^2 \text{ for } 1 < i \le n \\ \end{gather} \] where \( b^*_i = b^\prime_i - \sum_{j=1}^{i-1}\mu_{i,j}b^*_j \text{ and } \mu_{i,j} = \frac{ b^\prime_i \cdot b^*_j}{\lvert\lvert b^*_j \rvert\rvert^2} \)

To find an integer relation on \( x = (x_1, \dotsc, x_n) \in \RR^n \) we can use LLL on the lattice \[ L := \begin{bmatrix} 1 & 0 & 0 & \dotsm & 0 & Nx_1 \\ 0 & 1 & 0 & \dotsm & 0 & Nx_2 \\ \vdots & \vdots & \ddots & \dotsm & \vdots & \vdots \\ 0 & 0 & 0 & \dotsm & 1 & Nx_n \end{bmatrix}\] for some \( N \in \RR \).

Think of this as the augmented matrix \( \left[ I_n\, Nx \right] \).

Elements of the lattice look like \[ \left(\lambda_1, \dotsc, \lambda_n, N(\lambda_1x_1 + \dotsm + \lambda_nx_n) \right) \] So if the \( \lambda_i \) are an integer relation for \( x \) then the last term will be 0.

We can prove that if \( N \) is large enough, then the first element of the reduced lattice must contain the integer relation.

For the complex case where n \( z = (z_1, \dotsc, z_n) \in \CC^n \) we still use the augmented lattice \( L := \left[ I_n \, Nz \right] \) like we did for the real case.

However we then construct a 2nd augmented lattice \[ \begin{bmatrix} \Re{L} & \Im{L} \\ -\Im{L} & \Re{L}\end{bmatrix} \] and use LLL on that lattice.

As with the real case, so long as \( N \) is large enough then the procedure must find an integer relation.

In all cases, we don't know how large to make \( N \) before we start. So we must use trial and error.

We have found, experimentally, that with numeric (floating point) calculations it is possible for \( N \) to be too large, even though the theory requires \( N \) to be “sufficiently large”.

The PSLQ algorithm was introduced by Ferguson & Bailey in 1988. It was later analysed by Ferguson, Bailey, and Arno in 1999

It can handle both the real and complex cases directly.

We will not discuss the details of the algorithm itself. However we will establish some facts.

Observe that in both the real and complex cases \( \mathcal{O} \) forms a lattice, and that \( \mathcal{O} \subset \FF \).

Let \( x \in \FF \). An integer \( a \in \mathcal{O} \) is a *nearest integer* if \( \lvert x-a \rvert \) is maximal.
A function \( \lceil \cdot \rfloor : \FF \to \mathcal{O} \) is a nearest integer functino if it maps each \( x \in \FF \) to one of its nearest integers.

We denote by \( \lceil \cdot \rfloor_{\mathcal{O}} \) if we need to be specific.

There are parameters: \( \gamma, \rho \), and \( \tau \) that must satisfy \[ \begin{gather} \frac{1}{\rho} \ge \lvert x - \lceil x \rfloor \rvert \text{ for all } x \in \FF \\ 1 < \tau \le \rho \\ \frac{1}{\tau^2} = \frac{1}{\gamma^2} + \frac{1}{\rho^2} \end{gather} \]

\( \rho \) is bounded by the integer lattice; choose to minimise \(1/\rho\).

\( \gamma \) is minimised when \( \tau = 1 \); choose any larger value.

\( \tau \) is fixed by the choice of \( \gamma \) and the value of \( \rho \).

Each iteration of the PSLQ algorithm finds an increased lower bound on the norm of *any* integer relation on the input.

The algorithm can terminate when that bound exceeds some threshold.

If the algorithm finds a relation, \( a \) then \( \lvert\lvert a \rvert\rvert \le \gamma^{n-2}M \) where \( M \) is the norm of the smallest possible relation.

In Principle:

- We give PSLQ:
- The numbers we wish to find a relation for.
- An upper bound, \( b \), for the Euclidean norm of the relation we wish to find.

- PSLQ gives either:
- A relation, \( a \) where \( \norm{a} \le b \) and \( \norm{a} \le \gamma^{n-2}M \).
- A declaration that no such relation exists.

In Practice:

Due to floating point arithmetic and speed optimisations.

- We give PSLQ:
- The numbers we wish to find a relation for

- PSLQ gives:
- A relation accurate to the calculation precision with no bound guarantee.

- PSLQ is designed specifically for integer relation compuation
- In experimental testing:
- PSLQ required lower precision than LLL (for the same relation).
- PSLQ took less time than LLL (for the same relation).

Let \( \FF \) be a field. If \( \FF \) is a subfield of some other field \( \KK \) (i.e., \( \FF \subseteq \KK \) then we call \( \KK \) a *field extension* (or, equivalently an *extension field*) of \( \FF \).

We denote this by \( \KK:\FF \) when we need to be explicit about the base field and the extension.

Note that the literature often uses the notation \( \KK/\FF \) instead.

Let \( \FF \) be a field, and consider an extension field \( \KK:\FF \). An element \( k \in \KK \) is said to be *algebraic over \( \FF \)* if it satisfies
\[ f_nk^n + \dotsm + f_1 k + f_0 = 0 \]
for some \( n \in \NN \) and \( f_i \in \FF \). Note that \( f_n \ne 0 \).

For example \( \sqrt{2} \) satisfies \( k^2 - 2 = 0 \) and so is algebraic over the rationals.

Let \( D \) be an integral domain and consider an extension field \( \KK:\FF \) such that \( D \subseteq \KK \). An element \( k \in \KK \) is said to be *integeral over \( \FF \)* if it satisfies a monic polynomial
\[ k^n + d_{n-1}k^{n-1} + \dotsm + d_1 k + d_0 = 0 \]
for some \( n \in \NN \) and \( d_i \in D \)

An integral domain is a commutative ring with multiplicative identity and no non-zero elements \( d_1, d_2 \) with the property that \( d_1d_2 = 0 \).

Let \( z \in \CC \).

We denote by \( \QQ(z) \) the smallest subfield of \( \CC \) that contains both \( \QQ \) and \( z \).

We denote by \( \ZZ[z] \) the smallest subring of \( \CC \) that contains both \( \ZZ \) and \( z \).

Note that the ring \( \ZZ \) is often called the “rational integers” to avoid ambiguity.

We are primarily interested in extension fields of \( \QQ \).

A number \( z \in \CC \) is an *algebraic number* (or simply *algebraic*) if it is algebraic over \( \QQ \).

A number \( z \in \CC \) is an *algebraic integer* if it is integral over \( \ZZ \).

The ring of all algebraic integers is denoted by \( \mathcal{A} \).

An algebraic extension is *real* if \( \KK \subseteq \RR \) and *complex* otherwise.

We are primarily interested in extension fields of \( \QQ \).

An extension field \( \KK:\QQ \) is an *algebraic extension field* (or simply an *algebraic extension* if every \( k \in \KK \) is an algebraic number.

Its *ring of integers of \( \KK \)*, denoted \( \mathcal{O}_{\KK} \), is \( \KK \cap \mathcal{A} \). We call such rings *algebraic integers*

We will restrict our attention to quadratic extension fields. These are fields of the form:
\[ \QQ( \sqrt{D} ) = \left\{ q_1 + q_2 \sqrt{D} : q_i \in \QQ \right\} \]
wher \( D \in \ZZ \) is *square free*.

Elements of these fields satisfy *quadratic* (or lower degree) polynomials with rational coefficients.

The algebraic integers of a quadratic field (which we call *quadratic integers*) are of the form
\[ \mathcal{O}_{\QQ(\sqrt{D})} = \ZZ[\omega] = \left\{ m_1 + m_2 \omega : m_i \in \mathbb{Z} \right\} \]
where
\[
\omega = \begin{cases} \sqrt{D} & \text{ if } D \equiv 2,3 \;(\operatorname{mod} 4) \\ \frac{1+\sqrt{D}}{2} & \text{ if } D \equiv 1 \;(\operatorname{mod} 4) \end{cases}
\]

\( D \equiv 2,3 \;(\operatorname{mod} 4) \)

\( D \equiv 1 \;(\operatorname{mod} 4) \)

Finally we mention a special class of quadratic fields.

Recall the definiton of an integer relation:

For relations of algebraic integers we specify an intermediate extension field.

Let \( \FF \in \{ \RR, \CC \} \) and let \( \KK \) be an algebraic extension field. Let \( \mathcal{O} = \mathcal{O_{\KK}} \).

Let \( x_1, \dotsc, x_n \in \FF \). Integers, \( a_1, \dotsc, a_n \in \mathcal{O} \) are an *algebraic integer relation* (or a \( \KK \)-integer relation) of the original numbers \( x_i \) if
\[ a_1x_1 + \dotsm + a_nx_n = 0 \]

Observe that for an quadratic integer relation: \[ \begin{align*} & \left(m_{1,1} + m_{1,2} \omega\right)\,x_1 + \dotsm + \left(m_{n,1} + m_{n,2}\omega\right)\,x_n \\ =& \left(m_{1,1}x_1 + \dotsm + m_{n,1}x_n \right) + \left(m_{1,2}\, \omega x_1 + \dotsm m_{n,2}\,\omega x_n\right) \end{align*} \]

This suggests a way to reduce a quadratic integer relation to a classical one.

Given \( (x_1, \dotsc, x_n) \in \FF^n \) we construct \( (x_1, \dotsc, x_n, x_1\omega, \dotsc, x_n\omega) \in \FF^{2n} \) and search for a classical integer relation on that.

We recover the integer relation by taking \( a_k + a_{n+k}\omega \) for \( 1 \le k \le n \).

This works well for real quadratic fields (i.e., when \( D > 1 \)). It works less well for complex quadratic fields.

In the complex case, the integers are of the form \( a_k = m_{k,1} + m_{k,2} i \). So when we attempt to recover the relation we get \[ \left(m_{k,1} + m_{k,2} i\right) + \left(m_{k+n,1} + m_{k+n,2} i \right)\omega \] after expansion and some simplification we get \[ \alpha_1 + \beta_2 i \sqrt{\lvert D \rvert} + \alpha_2 i - \beta_2 \sqrt{\lvert D \rvert} \text{ for } \alpha_1, \beta_1 \in \ZZ \] which is not an algebraic integer in \( \QQ(\sqrt{D}) \) if \( \alpha_2 \ne 0 \) or \( \beta_2 \ne 0 \).

Much of the time the reduction technique works fine, and gives good answers even in the complex cases.

When it does not work, we try two techniques to nonetheless extract a relation.

Decomposition method

We observe that \[ \alpha_1 + \beta_2 i \sqrt{\lvert D \rvert} + \alpha_2 i - \beta_2 \sqrt{\lvert D \rvert}\] can be written as \[ \left( \alpha_1 + \beta_2 i\sqrt{\lvert D \rvert} \right) + i \left( \alpha_2 + \beta_2 i\sqrt{\lvert D \rvert} \right) \] and we have two possible algebraic integers embedded. We try extracting only these.

Conjugate Method

Given \[ \alpha_1 + \beta_2 i \sqrt{\lvert D \rvert} + \alpha_2 i - \beta_2 \sqrt{\lvert D \rvert} \] a conjugate is \[ \alpha_1 - \beta_2 i \sqrt{\lvert D \rvert} - \alpha_2 i - \beta_2 \sqrt{\lvert D \rvert} \]

Multiplying the number by its conjugate gives an element of \( \mathcal{O}_{\QQ(\sqrt{D})} \). We multiply all elements of the relation by the conjugate of only one of the elements.

Both the methods for correcting the complex case work well in practice.

We do not have proofs that they will always be correct. It is nonetheless easy to just try them, and check the extracted relation.

To compute an algebraic integer n \( z = (z_1, \dotsc, z_n) \in \FF^n \) we still use the augmented lattice \( L := \left[ I_n \, Nz \right] \) like we did for the real case.

However we then construct a 2nd augmented lattice. \[ \begin{bmatrix} \Re{L} & \Im{L} \\ \mathfrak{W_1}{L} & \mathfrak{W_2}{L}\end{bmatrix} \] and use LLL on that lattice.

However we then construct a 2nd augmented lattice. \[ \begin{bmatrix} \Re{L} & \Im{L} \\ \mathfrak{W_1}{L} & \mathfrak{W_2}{L}\end{bmatrix} \] and use LLL on that lattice.

\[ \mathfrak{W_1}{x_k} = \begin{cases} -\sqrt{\lvert D \rvert} \Im{x_k} & \text{ if } \text{ if } D \equiv 2,3 \;(\operatorname{mod} 4) \\ \left(\Re{x_k} - \sqrt{\lvert D \rvert}\Im{x_k} \right)/2 & \text{ if } D \equiv 1 \;(\operatorname{mod} 4) \end{cases} \] \[ \mathfrak{W_2}{x_k} = \begin{cases} \sqrt{\lvert D \rvert} \Re{x_k} & \text{ if } \text{ if } D \equiv 2,3 \;(\operatorname{mod} 4) \\ \left(\Im{x_k} + \sqrt{\lvert D \rvert}\Re{x_k} \right)/2 & \text{ if } D \equiv 1 \;(\operatorname{mod} 4) \end{cases} \]

As with the other cases, so long as \( N \) is large enough then the procedure must find an algebraic integer relation.

We attempt to modify PSLQ while keeping the existing theory.

We simply use a quadratic integer nearest integer function, and modify the algorithm to allow us to specify the quadratic extension field we wish to use.

We simply use a quadratic integer nearest integer function, and modify the algorithm to allow us to specify the quadratic extension field we wish to use.

This introduces a problem: for real quadratic extension fields, the integers do not form a lattice (there is no unique nearest integer to a field element), so we cannot use the existing theory.

We restrict our attention to the complex cases, then.

We restrict our attention to the complex cases, then.

In these cases we have a more subtle problem.

As \( D \) increases, the parameter \( \rho \) decreases, until eventually \( \rho < 1 \) making it impossible to satisfy the \( 1 < \tau \le \rho \) condition.

In these cases we have a more subtle problem.

As \( D \) increases, the parameter \( \rho \) decreases, until eventually \( \rho < 1 \) making it impossible to satisfy the \( 1 < \tau \le \rho \) condition.

The complex quadratic fields that do not exhibit these problems are exactly: \[ \QQ(\sqrt{-2}), \QQ(\sqrt{-3}), \QQ(\sqrt{-7}), \QQ(\sqrt{-11})\]

As \( D \) increases, the parameter \( \rho \) decreases, until eventually \( \rho < 1 \) making it impossible to satisfy the \( 1 < \tau \le \rho \) condition.

The complex quadratic fields that do not exhibit these problems are exactly: \[ \QQ(\sqrt{-2}), \QQ(\sqrt{-3}), \QQ(\sqrt{-7}), \QQ(\sqrt{-11})\]

These are precisely the complex quadratic norm Euclidean fields.

The complex quadratic fields that do not exhibit these problems are exactly: \[ \QQ(\sqrt{-2}), \QQ(\sqrt{-3}), \QQ(\sqrt{-7}), \QQ(\sqrt{-11}) \]

We note that we've had surprising success with the modified PSLQ on \[ \QQ(\sqrt{-5}), \QQ(\sqrt{-6}), \QQ(\sqrt{-10}) \] even though they do not fit the existing theory.

Thankyou

Are there any questions?

Matthew P. Skerritt