Generators of the Module of Syzygies

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"Time-Delay Interferometry"

Massimo Tinto and Sanjeev V. Dhurandhar

	Abstract
1	Introduction
2	Physical and Historical Motivations of TDI
3	Time-Delay Interferometry
4	Algebraic Approach for Canceling Laser and Optical Bench Noises
	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
5	Time-Delay Interferometry with Moving Spacecraft
	5.1	The unequal-arm Michelson
	5.2	The Sagnac combinations
	5.3	Algebraic approach to second-generation TDI
	5.4	Solutions with one arm nonfunctional
6	Optimal LISA Sensitivity
	6.1	General application
	6.2	Optimization of SNR for binaries with known direction but with unknown orientation of the orbital plane
7	Experimental Aspects of TDI
	7.1	Time-delays accuracies
	7.2	Clocks synchronization
	7.3	Clocks timing jitter
	7.4	Sampling reconstruction algorithm
	7.5	Data digitization and bit-accuracy requirement
8	Concluding Remarks
	Acknowledgement
A	Generators of the Module of Syzygies
B	Conversion between Generating Sets
	References
	Footnotes
	Updates
	Figures

A Generators of the Module of Syzygies

We require the 4-tuple solutions $(q3,q′1,q2′,q′3)$ to the equation

where for convenience we have substituted $x = 𝒟1$ , $y = 𝒟2$ , $z = 𝒟3$ . $q3$ , $q1′$ , $q′2$ , $q3′$ are polynomials in $x$ , $y$ , $z$ with integral coefficients, i.e., in $Z [x, y,z]$ .

We now follow the procedure in the book by Becker et al. [3 ].

Consider the ideal in $Z[x,y,z ]$ (or $𝒬 [x, y,z]$ where $𝒬$ denotes the field of rational numbers), formed by taking linear combinations of the coefficients in Eq. (114 *), $f1 = 1 − xyz$ , $f2 = xz − y$ , $f3 = x (1 − z2)$ , $f4 = 1 − x2$ . A Gröbner basis for this ideal is

The above Gröbner basis is obtained using the function GroebnerBasis in Mathematica. One can check that both the $fi$ , $i = 1,2,3, 4$ , and $gj$ , $j = 1,2,3$ , generate the same ideal because we can express one generating set in terms of the other and vice-versa:

where $d$ and $c$ are $4 × 3$ and $3 × 4$ polynomial matrices, respectively, and are given by

( ) − 1 − z2 − yz ( 2 ) | y 0 z | 0 0 − x z − 1 d = |( |) , c = ( − 1 − y 0 0 ) . (117 ) − x 0 2 0 0 z 1 0 − 1 − z − (x + yz)

The generators of the 4-tuple module are given by the set $A ⋃ B ∗$ , where $A$ and $B ∗$ are the sets described below:

$A$ is the set of row vectors of the matrix $I − d ⋅ c$ where the dot denotes the matrix product and $I$ is the identity matrix, $4 × 4$ in our case. Thus,

a1 = (z2 − 1,0,x − yz,1 − z2), 2 2 a2 = (0,z (1 − z ),xy − z,y (1 − z )), (118 ) a3 = (0,0, 1 − x2,x(z2 − 1)), 2 2 a4 = (− z ,xz, yz,z ).

We thus first get four generators. The additional generators are obtained by computing the S-polynomials of the Gröbner basis $𝒢$ . The S-polynomial of two polynomials $g1,g2$ is obtained by multiplying $g1$ and $g2$ by suitable terms and then adding, so that the highest terms cancel. For example in our case $g1 = z2 − 1$ and $g2 = y2 − 1$ , and the highest terms are $z2$ for $g1$ and $y2$ for $g2$ . Multiply $g1$ by $y2$ and $g2$ by $z2$ and subtract. Thus, the S-polynomial $p12$ of $g1$ and $g2$ is

Note that order is defined ( $x ≫ y ≫ z$ ) and the $y2z2$ term cancels. For the Gröbner basis of 3 elements we get 3 S-polynomials $p12$ , $p13$ , $p23$ . The $pij$ must now be re-expressed in terms of the Gröbner basis $𝒢$ . This gives a $3 × 3$ matrix $b$ . The final step is to transform to four-tuples by multiplying $b$ by the matrix $c$ to obtain $∗ b = b ⋅ c$ . The row vectors $∗ bi$ , $i = 1,2,3$ , of $∗ b$ form the set $∗ B$ :

b∗1 = (z2 − 1,y (z2 − 1),x (1 − y2),(y2 − 1)(z2 − 1)), ∗ 2 2 2 b2 = (0,z(1 − z ),1 − z − x (x − yz),(x − yz) (z − 1)) , (120 ) b∗3 = (− x + yz, z − xy,1 − y2,0).

Thus, we obtain three more generators, which gives us a total of seven generators of the required module of syzygies.