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lisa - laser interferometric space antenna - is a proposed esa - nasa mission which will use coherent laser beams exchanged between three identical spacecraft forming a giant ( almost ) equilateral triangle of side @xmath0 kilometres to observe and detect low frequency cosmic gw @xcite . laser frequency noise dominates the other secondary noises , such as optical path noise , acceleration noise by 7 or 8 orders of magnitude , and must be removed if lisa is to achieve the required sensitivity of @xmath1 , where @xmath2 is the metric perturbation caused by a gravitational wave . in lisa , six data streams arise from the exchange of laser beams between the three spacecraft approximately 5 million km apart . these six streams produce redundancy in the data which can be used to suppress the laser frequency noise by the technique called time - delay interferometry ( tdi ) in which the six data streams are combined with appropriate time - delays @xcite . a mathematical foundation for the tdi problem for the static lisa was given in @xcite , where it was shown that the data combinations cancelling laser frequency noise formed the _ module of syzygies _ over the polynomial ring of time - delay operators . for the static lisa , the polynomial ring was in three variables and also commutative . the generators of the module were obtained via grbner basis methods . this scheme can be extended in a straight forward way to include the ( sagnac ) effect arising from the rotation of lisa , where the up - down optical links are unequal and as a consequence , now one has six optical links , but the armlengths are still constant in time . these are the modified ist generation tdi . here the polynomial ring is still commutative although over six indeterminates . the generators of the module were obtained in @xcite . in the general case of time varying armlengths , the polynomial ring is in six variables ( the six optical links ) but now it is noncommutative . the algebraic problem in the general case has been discussed in @xcite , but the solution to the general problem seems extremely difficult . we still have a linear system which leads to a module ( a left module ) , but it seems difficult to obtain its generators in general . in the non - commutative case , even the grbner basis algorithm may not terminate . we must envisage the possibility that not all optical links of lisa will be operating at all times for various reasons like technical failure for instance or even the operating costs . therefore , it is important to discuss the question when not all the links operate . here we look at a specific situation where the data is available from only two arms or four optical links . this should not much affect the information that can be extracted from the data , because this is essentially a michelson configuration which is known to be quite useful . the practical advantage for this case is that the algebraic problem simplifies considerably and therefore becomes tractable . we can reduce the problem to that of only one linear constraint on two polynomials , although the equation is still noncommutative . we show that we can generate an infinity of solutions by using a combinatorial algebraic approach which lists all such solutions in a systematic way . the solutions - that is the laser frequency noise is suppressed for these time - delayed combinations - are approximate in the sense that the @xmath3 and @xmath4 terms are ignored in the calculation , where @xmath5 is the generic length of the lisa s arm or the optical link and the dot denotes derivative with respect to time . the solutions are based on vanishing commutators ( in this approximation ) . we enumerate such commutators and for each such commutator there is a corresponding solution . we present an algorithm to construct these solutions . a geometric combinatorial approach was adopted in @xcite where several solutions were exhibited . our approach is algebraic where several operations are algebraic operations on strings of operators . the algebraic approach has the advantage of easy manipulation of data streams , although some geometrical insight could be at a premium . the analysis presented here can be used also for other future space detectors like astrod where again a similar situation arises @xcite . in this section we recapitulate the results obtained in the earlier literature already mentioned and also set up notation . in the literature on tdi of lisa there are several notations - here we choose the one which seems simplest for our purpose . we follow the notation and conventions of @xcite and @xcite . the six links are denoted by @xmath6 . the time - delay operator for the link @xmath7 from s / c 1 to s / c 2 or @xmath8 is denoted by @xmath9 in @xcite and so on in a cyclic fashion . the delay operators in the other sense are denoted by @xmath10 ; the link @xmath11 from @xmath12 by @xmath13 and similarly the links @xmath14 are defined through cyclic permutation . figure [ schematic ] depicts the optical links as described . let @xmath15 represent the laser frequency noise in s / c @xmath16 . let @xmath17 be the delay operator corresponding to the variable armlength @xmath18 , i.e. @xmath19 . then we have , u^1 = c_1 - z c_3 , + v^1 = l c_2 - c_1 . the other links in terms of @xmath20 are obtained by cyclic permutations . also in the @xmath21 we have not included contributions from the secondary noises , gravitational wave signal etc . since here our aim is to deal with laser frequency noise only . any observable @xmath22 is written as : x = p_i v^i + q_i u^i , where @xmath23 are polynomials in the variables @xmath24 . thus @xmath22 is specified by giving the six tuple polynomial vector @xmath25 . writing out the @xmath26 in terms of the laser noises @xmath20 , and in order that the laser frequency noise cancel for arbitrary functions @xmath20 , the polynomials @xmath25 must satisfy the equations : p_1 - q_1 + q_2 x - p_3 n & = & 0 , + p_2 - q_2 + q_3 y - p_1 l & = & 0 , + p_3 - q_3 + q_1 z - p_2 m & = & 0 , [ lneq ] where one must remember that the order of the operators is important when the armlengths are time - varying . the solutions ( if they exist ) to eq . ( [ lneq ] ) are the second generation tdi . one finds that at least in the approximation we consider they do exist . for a reasonably optimised model of lisa , say given in @xcite , @xmath27 metres / sec@xmath28 and thus even if one considers say 20 successive optical paths , that is , about @xmath29 seconds of light travel time , @xmath30 metre . this is well below few metres and thus can be neglected in the residual laser noise computation . moreover , @xmath31 terms ( and higher order ) can be dropped since they are of the order of @xmath32 ( they come with a factor @xmath33 ) which is much smaller than 1 part in @xmath34 , which is the level at which the laser frequency noise must be cancelled . thus , we keep terms only to the first degree in @xmath35 and also neglect higher time derivative terms in @xmath36 . the solutions to these equations are the polynomial vectors @xmath37 and they form a left module @xmath38 over the non - commutative ring @xmath39 , where @xmath40 is the field of rational numbers and @xmath41 are the indeterminates . a left module means that one can multiply a solution @xmath25 from the _ left _ by any polynomial in @xmath42 , then it is also a solution to the equations ( [ lneq ] ) and therefore in the module @xmath38 . it is difficult to solve these equations in general for the non - commutative case . previously we have shown in @xcite that the ring @xmath42 can be reduced to the quotient ring @xmath43 which is defined by quotienting @xmath42 by the ideal @xmath44 generated by vanishing commutators . consider commutators of the form @xmath45 $ ] where @xmath46 and @xmath47 or @xmath48 represents any one of the delay operators @xmath41 . upto the order of approximation we work in : = _ k = 1^n l_x_k _ m = 1^n _ y_m - _ m = 1^n l_y_m _ k = 1^n _ x_k . from this equation it immediately follows that if the set of operators @xmath49 is a permutation of the operators @xmath50 , then the commutator : = 0 , [ xct_cmtr ] upto the order of the approximation . the ideal @xmath44 is generated by such commutators . although we may be able to apply non - commutative grbner basis methods , the general solution seems quite difficult . however , we could consider a simpler problem of one arm of lisa being nonfunctional . then we have a simpler situation which turns out to be tractable . we arbitrarily choose the non - functional arm to be the one connecting s / c 2 and s / c 3 . this means from our labelling that the polynomials are now restricted to only the four variables @xmath51 and we can put the polynomials @xmath52 . this simplifies the equations ( [ lneq ] ) considerably . the second and third of the equations in ( [ lneq ] ) simplify to : q_2 & = & - p_1 l , + p_3 & = & - q_1 z . [ eqsimp ] substituting these in the first equation , we need to solve just one equation : p_1 ( 1 - lx ) - q_1 ( 1 - zn ) = 0 , to obtain the full polynomial vector as a solution . it is clear that solutions are of the michelson type denoted by @xmath22 . one such solution has already been given in the literature @xcite . this solution in our notation is : p_1 & = & 1 - zn - znlx + lx ( zn)^2 + q_1 & = & 1 - lx - lxzn + zn ( lx)^2 . [ x1 ] writing , = p_1 ( 1 - lx ) - q_1 ( 1 - zn ) , we get for ( [ x1 ] ) , @xmath53 $ ] which is a commutator that vanishes . thus it is an element of @xmath44 and ( [ x1 ] ) is a solution ( over the quotient ring ) . this is already well known . _ what we would like to emphasise is that there are more solutions of this type - in fact there are infinite number of such solutions . _ in what follows we give a systematic way to enumerate and list such solutions . notice that the operators @xmath54 and @xmath55 occur together . physically , these operators correspond to round trips . we denote these round trip operators by @xmath56 and @xmath57 , then we can write the previous equation and solution respectively , in a compact way as : p_1 ( 1 - a ) - q_1 ( 1 - b ) = 0 , [ 4lnk ] and , p_1 & = & 1 - b - ba + ab^2 , + q_1 & = & 1 - a - ab + ba^2 . [ x1 ] we can substitute this in @xmath58 to obtain @xmath59 $ ] which vanishes and hence ( [ x1 ] ) is a solution . but we can proceed further and write : p_1 & = & p_1 - b a^2 b , + q_1 & = & q_1 - a b^2 a , to obtain :  = b a^2 ba - a b^2 ab . @xmath60 however does not vanish - it is also not a commutator - and hence @xmath61 is not a solution . continuing further with , p_1  & = & p_1 + a b^2 a b , + q_1  & = & q_1 + b a^2 b a , we get , @xmath62 which also does not vanish , again not a commutator . this process can be continued in such a way so that lower degree terms cancel and so as to get another vanishing commutator but of higher degree . the next commutator is obtained by adding two more terms of successively higher degree in a similar way . the result is : p_1 & = & 1 - b - ba + ab^2 - b a^2 b + a b^2 ab + ab^2 aba - ba^2 bab^2 , + q_1 & = & 1 - a - ab + ba^2 - a b^2 a + b a^2 ba + ba^2 bab - ab^2 aba^2 . [ x2 ] note that @xmath63 is obtained from @xmath64 by interchanging @xmath65 and @xmath66 and vice versa . this is a symmetry . the expressions for @xmath64 and @xmath63 in eq . ( [ x2 ] ) give the vanishing commutator : = [ a b^2 a , b a^2 b ] , and hence ( [ x2 ] ) is a solution . there are two more solutions with eight terms . one of them is : p_1 & = & 1 + a - b^2 - b^2 a - b^2 a^2 - b^2 a^3 + a^2 b^4 + a^2 b^4 a + & = & ( 1 - b^2 - b^2 a^2 + a^2 b^4)(1 + a ) , + q_1 & = & 1 + b - a^2 - a^2 b - a^2 b^2 - a^2 b^3 + b^2 a^4 + b^2 a^4 b + & = & ( 1 - a^2 - a^2 b^2 + b^2 a^4)(1 + b ) , with @xmath67 $ ] which is also a vanishing commutator . this solution looks something like the square of the solution ( [ x1 ] ) obtained by tinto et al ( see @xcite ) multiplied by linear factors in order ensure successive term cancellation . here one starts with @xmath68 and continues until one reaches the above commutator . the third solution corresponds to the commutator @xmath69 $ ] . this solution is given by : p_1 & = & 1 - b + ab - bab - baba + abab^2 - baba^2 b + abab^2 ab , + q_1 & = & 1 - a + ba - aba - abab + baba^2 - abab^2 a + baba^2 ba . the following observations are in order : the solutions correspond to the vanishing commutators as follows ; there is only one commutator at degree 4 , namely , @xmath70 $ ] ; of degree 8 , there are three commutators : @xmath71,~ [ abab , baba],~ [ a b^2 a , b a^2 b]$ ] . and these are the only three commutators at degree 8 . also there are equal number of @xmath65 s and @xmath66 s in each string of the commutators . to each of the vanishing commutators , there is a corresponding polynomial vector solution @xmath72 . the rest of the solution vector is obtained from eq.([eqsimp ] ) which then yields @xmath73 and @xmath74 . these are solutions only upto degree 7 . clearly one can construct solutions of higher degree . we do this in a combinatorial way in the next section . we enumerate the solutions in a given order - called the lexicographic order - and also present an algorithm to obtain the polynomials @xmath64 and @xmath63 . in order to proceed further , let us first define an interchange operator @xmath75 between the symbols @xmath65 and @xmath66 , that is , given a string @xmath76 composed of @xmath65 and @xmath66 , the string @xmath77 is obtained by replacing @xmath65 by @xmath66 in @xmath76 . a string is a polynomial with only one term usually also called a monomial in the literature . the length of a string is the degree of the monomial . we will use these terms interchangably , but that should not lead to any confusion . note that all the vanishing commutators which have appeared corresponding to a solution have the form @xmath78 $ ] . also since the string @xmath79 must be a permutation of the string @xmath76 , in order that the commutator @xmath78 $ ] vanish , there must be an equal number of @xmath65 s and @xmath66 s in the string @xmath76 of the vanishing commutator . so if we have @xmath80 number of @xmath65 s in the string @xmath76 , it must also have the same @xmath80 number of @xmath66 s , so that the total string @xmath76 has length @xmath81 . the commutator is then made up of strings of length @xmath82 . note that the operator @xmath75 can be extended to a polynomial containing strings in an obvious way by defining : i ( s_1 + s_2 + ... ) = i s_1 + i s_2 + ... . then we easily check that in the foregoing @xmath83 , where @xmath72 constitute a solution to eq . ( [ 4lnk ] ) . also @xmath84 the identity mapping . @xmath75 can be considered to be an operator over the polynomial ring over the indeterminates @xmath65 and @xmath66 .    in the foregoing we have then considered cases for @xmath85 . for @xmath86 there is one solution , while for @xmath87 there are three distinct solutions . in order to proceed further , we must first list the commutators at each @xmath80 . in order to carry out this programme , we must introduce an order among the strings of length @xmath81 . we choose this order to be _ lexicographic _ or if we consider strings of all lengths , then _ length lexicographic _ @xcite . but since at a time we deal only with strings of a given length @xmath81 , the lexicographic order suffices . we arbitrarily choose @xmath88 . then at length 2 , @xmath89 . at length 3 , @xmath90 and so on . more generally , for strings of the same length , @xmath91 if at the first position from the left the strings differ , the symbol in string @xmath76 is less than the symbol in string @xmath92 . in this way we may list all strings in the symbols @xmath93 of a given length this is all we require . thus at length @xmath94 we have just one commutator @xmath95 $ ] . at @xmath96 , we have @xmath97 , [ abab , baba ] , [ abba , baab]$ ] , that is , three commutators ; we have also three solutions corresponding to each commutator . we have also taken care to write the first string in the commutator to be less than the second string in the lexicigraphic order chosen . note that we could list the commutator just by the first string , because the second string in the commutator is obtained by applying the operator @xmath75 to the first string . we will give an algorithm to obtain a solution , that is , the polynomials @xmath64 and @xmath63 in a unique way given the commutator . thus listing all the commutators of a string of length @xmath81 amounts to listing the solutions @xmath72 having that commutator . the polynomials @xmath64 and @xmath63 have @xmath81 terms and degree @xmath98 . we are now ready to list the commutators @xmath78 $ ] with first string @xmath76 of length @xmath81 . consider @xmath99 . the string @xmath76 has length 6 with 3 @xmath65 s and 3 @xmath66 s . listing the strings in lexicographical order we have : @xmath100 . we stop here because , the next string is @xmath101 and @xmath102 has occurred in the listing before and hence would lead to minus the commutator and therefore not an independent commutator . this follows from the symmetry encoded in the interchange operator @xmath75 . thus at @xmath99 there are 10 independent commutators . proceeding in this way we find that at @xmath103 there are 35 such commutators . the general formula for the number of commutators is @xmath104 . this formula follows from the following argument . consider a string of length @xmath81 , with @xmath80 number of @xmath65 s and @xmath66 s . because of symmetry between @xmath65 and @xmath66 , the first @xmath65 is fixed in the string ; as seen , the string starting with @xmath66 does not lead to a new commutator . thus we must choose @xmath105 combinations of @xmath65 s among @xmath98 available positions in the remaining string . this is the number of independent commutators which is then just @xmath104 . the algorithm is obtained by observing the way we got the solutions of degree 7 for @xmath87 . we started with the lowest degree and proceeded towards obtaining the commutator by cancelling terms at successively increasing degree . but now we have been assigned a commutator before hand and we need to find the solutions @xmath72 . we therefore need to go in the reverse order . we freely make use of the operator @xmath75 in this algorithm . consider the commutator @xmath106 = s_{2n } i ( s_{2n } ) - i ( s_{2n } ) s_{2n}$ ] where @xmath107 is a string containing @xmath80 number of @xmath65 s and an equal number of @xmath66 s .    1 . consider the first string in the commutator @xmath108 which we denote by @xmath109 . the subscript denotes the length of the string or the degree of the monomial . now , either @xmath110 ends in @xmath65 or @xmath66 ; that is , @xmath111 or @xmath112 . * if @xmath111 , then write @xmath113 or ; * if @xmath112 , then write @xmath114 . 2 .   replace @xmath110 by @xmath115 and repeat . that is either we have @xmath116 or @xmath117 . if @xmath116 , then write @xmath118 otherwise write @xmath119 . 3 .   continue this process till one arrives at a string of length 0 , namely , @xmath120 . note that @xmath121 . 4 .   then @xmath122 and @xmath83 . note that @xmath123 are strings of length @xmath124 . @xmath64 and @xmath63 are polynomials of degree @xmath125 .    to illustrate the algorithm explicitly , let us take the case for @xmath99 and choose one commutator among the 10 possible ones . let us randomly choose the string @xmath126 , so that the commutator is @xmath127 $ ] . so we have @xmath128 . thus @xmath129 . now @xmath130 ends in @xmath66 . thus we write @xmath131 , where @xmath132 . thus choosing the second option in step ( ii ) , we have @xmath133 . this is the highest degree term in the polynomial @xmath64 and for @xmath99 , it is of degree 11 . the next term is obtained from @xmath134 . again @xmath134 ends in @xmath66 . following the steps of the algorithm , @xmath135 and hence @xmath136 . since , @xmath137 ends in @xmath65 , @xmath138 . we then have @xmath139 and so on . we must carry out the steps till we reach degree 0 . this terminates the algorithm . in this way , we can easily construct the 11th degree polynomial @xmath64 by adding up all the strings @xmath140 . collecting all the terms and writing out the polynomial @xmath64 explicitly : p_1 & = & 1 - b + ab - bab - baba + abab^2 - baba^2b + abab^2ab + & - & baba^2bab + abab^2abab + abab^2ababa - baba^2babab^2 .    the polynomial @xmath63 is obtained from @xmath64 by just interchanging @xmath65 and @xmath66 , that is , @xmath83 . the other polynomials @xmath74 and @xmath73 are obtained from eq . ( [ eqsimp ] ) . these polynomials then constitute the polynomial vector which is a solution and is in the module of solutions over the quotient ring @xmath43 . in terms of the time - delay operators @xmath51 , the polynomials @xmath64 and @xmath63 are of degree 22 , while @xmath74 and @xmath73 are polynomials of degree 23 . for a general value of @xmath80 , the solution contains polynomials of maximum degree @xmath141 in the time - delay operators . clearly , from the mathematical point of view we have an infinite family of solutions . note that here we do not lay any claim to any exhaustive listing of solutions . however , we believe this family of solutions is sufficiently rich , because we can form linear combinations of these solutions and then these form a left submodule of the full left - module of solutions . we can also subtract solutions like ( [ x1 ] ) from ( [ x2 ] ) which is also a solution with each component polynomial having less number of terms ( or links in the usual terminology ) albeit of the same degree . further , by using cyclic permutations on @xmath142 and @xmath143 we can generate more solutions . for example , we may set @xmath144 and @xmath57 and obtain solutions for the four links , @xmath145 . the linear combinations of these solutions with the previous ones in the variables @xmath146 are also solutions in the full module pertaining to all six links . thus we see that a plethora of solutions can be so generated .    from the physical point of view , since we have neglected terms in @xmath3 and @xmath4 and higher orders in our calculations , a limit on the degree of the polynomial solutions arises . that is upto certain degree of the polynomials , we can safely assume the commutators to vanish . but as the degree of the polynomials increases it is not possible to neglect these higher order terms and then such a limit becomes important . we now investigate this limit and make a very rough estimate of it . our lisa model in @xcite gives @xmath147 m / sec@xmath28 . the other term @xmath148 is way too small to matter . so the limit is set by @xmath149 . from @xmath149 we compute the error in @xmath36 , namely , @xmath150 . if we allow the error to be no more than say 10 metres , then we find @xmath151 sec . each time - delay is about @xmath152 secs which gives the number of successive time - delays to be about 270 . this is the maximum degree of the polynomials . this means one can go upto @xmath153 . if we set the limit more stringently at @xmath154 metre , then the highest degree of the polynomial reduces to about 80 which means one can go upto @xmath155 . this means there are a large number of tdi observables available to do the physics . we however point out that when we go to higher degree polynomials , the number of interpolations of data also increases leading to accumulation of numerical errors . this is a relevant issue whenever high degree polynomials are used . in this work we have given a family of approximate solutions for the general model of lisa where the up - down links are unequal and the armlengths are time varying , but with only two arms functional . such a situation can arise when one of the arms is non - functional or when it is not possible to obtain data from all optical links because high costs etc . we then have only four links to deal with and although the general non - commutative problem of obtaining second generation tdi is extremely difficult , the problem here reduces to a much simpler and tractable one . we have therefore been able to exhibit a family of solutions although not exhaustive , is quite rich and interesting both from the mathematical and physical point of view . also the mathematical analysis presented here could be extended to other space missions such as astrod where a similar situation arises although the physical parameters will be quite different . the authors s. v. dhurandhar and j - y vinet would like to thank the indo - french centre for the promotion of advanced research ( ifcpar ) project no . 3504 - 1 under which this work has been carried out .          armstrong j w 2008 _ lrr-2006 - 1 : http://relativity.livingreviews.org / articles / lrr-2008 - 2_. s. v. dhurandhar , k. rajesh nayak , j - y . vinet , _ phys . * d 65 * , 102002(2002 ) . k. r. nayak and j - y vinet , _ phys . _ d * 70 * , 102003 ( 2004 ) . s. v. dhurandhar , jpcs , * 154 * , 012047 ( 2009 ) . m. vallisneri , _ phys . _ d * 72 * , 04003 ( 2005 ) . et al , advances in space research , * 32 * , 1437 ( 2003 ) . s. v. dhurandhar , j - y . vinet and r. nayak , class . quantum grav . * 25 * , 245002 ( 2008 ) . b. j. keller , _ algorithms and orders for finding noncommutative grbner basis _ , thesis , viginia 1997 .
in order to attain the requisite sensitivity for lisa - a joint space mission of the esa and nasa- the laser frequency noise must be suppressed below the secondary noises such as the optical path noise , acceleration noise etc . by combining six 
 appropriately time - delayed data streams containing fractional doppler shifts - a technique called time delay interferometry ( tdi ) - the laser frequency noise may be adequately suppressed . 
 we consider the general model of lisa where the armlengths vary with time , so that second generation tdi are relevant . 
 however , we must envisage the possibility , that not all the optical links of lisa will be operating at all times , and therefore , we here consider the case of lisa operating with two arms only . as shown earlier in the literature , 
 obtaining even approximate solutions of tdi to the general problem is very difficult . since here 
 only four optical links are relevant , the algebraic problem simplifies considerably . 
 we are then able to exhibit a large number of solutions ( from mathematical point of view an infinite number ) and further present an algorithm to generate these solutions .