We first consider the simpler case, where we ignore the optical-bench motion noise and consider only the laser phase noise. We do this because the algebra is somewhat simpler and the method is easy to apply. The simplification amounts to physically considering each spacecraft rigidly carrying the assembly of lasers, beam-splitters, and photodetectors. The two lasers on each spacecraft could be considered to be locked, so effectively there would be only one laser on each spacecraft. This mathematically amounts to setting and . The scheme we describe here for laser phase noise can be extended in a straight-forward way to include optical bench motion noise, which we address in the last part of this section.

The data combinations, when only the laser phase noise is considered, consist of the six suitably delayed
data streams (inter-spacecraft), the delays being integer multiples of the light travel times between
spacecraft, which can be conveniently expressed in terms of polynomials in the three delay operators ,
, . The laser noise cancellation condition puts three constraints on the six polynomials of the delay
operators corresponding to the six data streams. The problem therefore consists of finding six-tuples of
polynomials which satisfy the laser noise cancellation constraints. These polynomial tuples form a
module^{1}
called the module of syzygies. There exist standard methods for obtaining the module, by which we mean
methods for obtaining the generators of the module so that the linear combinations of the generators
generate the entire module. The procedure first consists of obtaining a Gröbner basis for the ideal
generated by the coefficients appearing in the constraints. This ideal is in the polynomial ring in the
variables , , over the domain of rational numbers (or integers if one gets rid of the
denominators). To obtain the Gröbner basis for the ideal, one may use the Buchberger algorithm or use
an application such as Mathematica [35]. From the Gröbner basis there is a standard way
to obtain a generating set for the required module. This procedure has been described in the
literature [2, 16]. We thus obtain seven generators for the module. However, the method does not
guarantee a minimal set and we find that a generating set of 4 polynomial six-tuples suffice to
generate the required module. Alternatively, we can obtain generating sets by using the software
Macaulay 2.
The importance of obtaining more data combinations is evident: They provide the necessary
redundancy - different data combinations produce different transfer functions for GWs and the
system noises so specific data combinations could be optimal for given astrophysical source
parameters in the context of maximizing SNR, detection probability, improving parameter estimates,
etc.

4.1 Cancellation of laser phase noise

4.2 Cancellation of laser phase noise in the unequal-arm interferometer

4.3 The module of syzygies

4.4 Gröbner basis

4.5 Generating set for the module of syzygies

4.6 Canceling optical bench motion noise

4.7 Physical interpretation of the TDI combinations

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