Rational Invariants Of Finite Abelian Groups
Rational Invariants of Finite Abelian GroupsGeorge LabahnSymbolic Computation GroupCheriton School of Computer ScienceUniversity of Waterloo, Canada.* Joint work with Evelyne Hubert, INRIA Méditerranée.Kolchin Seminar : March 11, 2016George LabahnRational Invariants of Finite Abelian Groups1/37
Rational Invariants of Finite Abelian Groups(1) Given finite, abelian group G GL(n, K) acting on KnGeorge LabahnRational Invariants of Finite Abelian Groups2/37
Rational Invariants of Finite Abelian Groups(1) Given finite, abelian group G GL(n, K) acting on Kn- construct rational invariants of action rational invariant : f K(x) : f ( g · x ) f ( x ) g G- determine rewrite rules for this actionGeorge LabahnRational Invariants of Finite Abelian Groups2/37
Rational Invariants of Finite Abelian Groups(1) Given finite, abelian group G GL(n, K) acting on Kn- construct rational invariants of action rational invariant : f K(x) : f ( g · x ) f ( x ) g G- determine rewrite rules for this action(2) Given system of polynomial equations- if have group action then ‘reduce’ polynomial system- conversely : determine finite abelian group action(if possible)George LabahnRational Invariants of Finite Abelian Groups2/37
ReferencesThis talk is a report on paperHubert & Labahn, Rational invariants of Finite Abelian Groups,To appear in Mathematics of ComputationRelevant other publications- K. Gatermann (ISSAC 1990): Using group actions to reduce Gröbner bases comp.- J-C Faugère and J. Svartz (ISSAC 2013): Using abelian group actions to reduce polynomial systems.- E. Hubert and G. Labahn (ISSAC 2012, FoCM 2013): Scaling symmetriesGeorge LabahnRational Invariants of Finite Abelian Groups3/37
Example : Invariant Polynomial SystemConsider the following system of polynomial equationsx1 x2 x3 x1 x2 x1 x3 x2 x3 12x1 x2 x2 x3 x1 x3 15x1 x2 x3 13George Labahn 0 0 0Rational Invariants of Finite Abelian Groups4/37
Example : Invariant Polynomial SystemConsider the following system of polynomial equationsx1 x2 x3 x1 x2 x1 x3 x2 x3 12x1 x2 x2 x3 x1 x3 15x1 x2 x3 13 0 0 0Solution space of system is invariant under the order 3 permutation(x1 , x2 , x3 ) (x2 , x3 , x1 ).Goal : work “modulo” this order 3 permutation.George LabahnRational Invariants of Finite Abelian Groups4/37
Example : Invariant Polynomial System(i) Find invariants 3 x 2 x 3 x x 2 3 x 2 x η2 3 x 2 x 3 x x 2 3 x x 2 η x 3 6 x x x x 3 x 3 1 31 22 311 2 3232 3 1 2 1 3 y1 2 2 x 2 x 2yxx xx xxη xx xx xxη x 1 21 32 31 21 32 3123 2 y3x1 x2 x3( η primitive cube root of unity) . (ii) Rewrite system in terms of invariants3y2 3y3 3y3 2 12 3y2 3y3 2 15y3y1 2 y3 3 3y2 y3 13y1 0, 0, 0.(iii) Solve invariant system for (y1 , y2 , y3 )(2 solutions)(iv) Work back to get (x1 , x2 , x3 )(6 solutions)George LabahnRational Invariants of Finite Abelian Groups5/37
The ProcessChange coordinatesScaling actionsArithmetic with exponentsPolynomial system : solve and work backGeorge LabahnRational Invariants of Finite Abelian Groups6/37
The Process : Z3- Change to ‘Fourier’ coordinates (x1 , x2 , x3 ) (z1 , z2 , z3 ): η η2 1 z1 x1 2 η 1 · z2 x2 η 1 1 1x3z3where η is a primitive cube root of unity.George LabahnRational Invariants of Finite Abelian Groups7/37
The Process : Z3- Change to ‘Fourier’ coordinates (x1 , x2 , x3 ) (z1 , z2 , z3 ): η η2 1 z1 x1 2 η 1 · z2 x2 η 1 1 1x3z3where η is a primitive cube root of unity.- Polynomial system now looks like :0 3z1 z2 3z3 3z23 120 3z1 z2 3z23 150 z31 z32 z33 3z1 z2 z3 13George LabahnRational Invariants of Finite Abelian Groups7/37
The Process : Z3- Change to ‘Fourier’ coordinates (x1 , x2 , x3 ) (z1 , z2 , z3 ): η η2 1 z1 x1 2 η 1 · z2 x2 η 1 1 1x3z3where η is a primitive cube root of unity.- Group action now looks like :(z1 , z2 , z3 ) ( η · z1 , η2 · z2 , z3 )Note : Action looks like ‘rescaling’ of coordinatesGeorge LabahnRational Invariants of Finite Abelian Groups8/37
The Process : Z3- Change to ‘Fourier’ coordinates (x1 , x2 , x3 ) (z1 , z2 , z3 ): η η2 1 z1 x1 2 η 1 · z2 x2 η z31 1 1x3where η is a primitive 3 root of unity.- Group action now looks like :(z1 , z2 , z3 ) ( η · z1 , η2 · z2 , z3 )Notice :George Labahnz2,z21z32 ,z1 z2 z3 all rational invariant functionsRational Invariants of Finite Abelian Groups9/37
The Process : Z3- Change to ‘Fourier’ coordinates (x1 , x2 , x3 ) (z1 , z2 , z3 ): x1 η η2 1 z1 2 η 1 · z2 x2 η x31 1 1z3where η is a primitive cube root of unity.- Group action now looks like :(z1 , z2 , z3 ) ( η · z1 , η2 · z2 , z3 )Notice : z31 ,George Labahnz1 · z2 ,z3 all rational invariant functionsRational Invariants of Finite Abelian Groups10/37
The Process : Z3- Change to ‘Fourier’ coordinates (x1 , x2 , x3 ) (z1 , z2 , z3 ): x1 η η2 1 z1 2 η 1 · z2 x2 η x31 1 1z3where η is a primitive cube root of unity.- Group action now looks like :(z1 , z2 , z3 ) ( η · z1 , η2 · z2 , z3 )Notice : z31 ,z1 · z2 ,z3 all rational invariant functionsHow to rewrite polynomial system in terms of invariants?George LabahnRational Invariants of Finite Abelian Groups10/37
The Process : Z3- Group action now looks like :(z1 , z2 , z3 ) ( η · z1 , η2 · z2 , z3 )Notice :y za1 · zb2 · zc3 ηa 2b za1 · zb2 · zc3is a rational invariant function iffa 2b 0mod 3- Kernel determined via integer linear algebra on exponents.- Rewrite rules reverse such kernel operationsGeorge LabahnRational Invariants of Finite Abelian Groups11/37
The Process : Z3Original transformed polynomial system:0 3z1 z2 3z3 3z23 120 3z1 z2 3z23 150 z31 z32 z33 3z1 z2 z3 13 y1 z31 , y2 z1 · z2 , y3 z3 z1 y1/3, z2 1y2,y1/31 rational invariantsz3 y3 rewrite rules- Rational invariant system0 3y2 3y3 3y23 120 3y2 3y23 150 y1 George Labahny2 3 y33 3y2 y3 13y1Rational Invariants of Finite Abelian Groups12/37
General Process : G GLn (K)(1) Fourier stepmatrix diagonalization (2) Finite group diagonalization scaling- order and exponent matrices(3) Rational invariants kernel of exponents order- integer linear algebra(4) Rewrite rulesGeorge Labahn ‘inverting’ kernelsRational Invariants of Finite Abelian Groups13/37
‘Fourier Step Diagonalization ’G : finite abelian subgroup GLn (K)George Labahn(order p p1 · · · ps )Rational Invariants of Finite Abelian Groups14/37
‘Fourier Step Diagonalization ’G : finite abelian subgroup GLn (K)(order p p1 · · · ps )(i) G is diagonalizable .- matrix R such that D R 1 · G · R all diagonal matricesGeorge LabahnRational Invariants of Finite Abelian Groups14/37
‘Fourier Step Diagonalization ’G : finite abelian subgroup GLn (K)(order p p1 · · · ps )(i) G is diagonalizable .- matrix R such that D R 1 · G · R all diagonal matrices- Letx R · z. Then have diagonal actionD Kn Kn( diag( d1 , . . . , dn ), (z1 , . . . , zn ) ) 7 (d1 · z1 , . . . , dn · zn )George LabahnRational Invariants of Finite Abelian Groups14/37
‘Exponents Finite Direct Sum ’G : finite abelian subgroup GLn (K)(order p p1 · · · ps )(ii) Group isomorphism : D Zp1 . . . ZpsExplicit via exponents :Zp1 . . . Zps(m1 , . . . , ms )George Labahn Dms17 Dm1 · · · DsRational Invariants of Finite Abelian Groups15/37
‘Exponents Finite Direct Sum ’G : finite abelian subgroup GLn (K)(order p p1 · · · ps )(ii) Group isomorphism : D Zp1 . . . ZpsExplicit via exponents :Zp1 . . . Zps Dms17 Dm1 · · · Ds(m1 , . . . , ms )Diagonal action:( diag( d1 , . . . , dn ), (z1 , . . . , zn ) ) 7 (d1 · z1 , . . . , dn · zn )with eachGeorge Labahnmmdj D1 1j · · · Ds sjRational Invariants of Finite Abelian Groups15/37
NotationDiagonal action : (α, β) Z7 Z5 :!βα3 3(z1 , z2 , z3 , z4 , z5 ) α z1 , β z2 , 4 z3 , 4 z4 , α β z5 .αβ63Exponent and Order matrices:"#6 0 4 1 3A : 0 3 1 4 3P : "75#Notation:β α( α, β ) α , β , 4 , 4 , α3 β3α βA63!(z1 , z2 , z3 , z4 , z5 ) ( α, β )A (z1 , z2 , z3 , z4 , z5 )George LabahnRational Invariants of Finite Abelian Groups16/37
Finite Abelian Group ActionsRational invariantsInteger linear algebraRewrite rulesGeorge LabahnRational Invariants of Finite Abelian Groups17/37
Rational Invariants K(z)AF(z) is invariant under z 7 λA z ifF(λA z) F(z)LemmaLaurent monomials: zv zv11 · · · zvnn , v Zn . Invariant iff(λA z)v zv A · v 0 mod PLemmaRational Invariants: F(z) K(z)A :XF(z) v kerZ AXv kerZ AGeorge Labahnmod Pmod Pav zvbv zvRational Invariants of Finite Abelian Groups18/37
Kernel? Use Hermite Normal FormDiagonal action : (α, β) Z6 Z3 :"4 1 1 34 3 6 00 3#[ A, P ]exponentmatrixGeorge LabahnRational Invariants of Finite Abelian Groups19/37
Kernel? Use Hermite Normal FormDiagonal action : (α, β) Z6 Z3 :"4 1 1 34 3 6 00 3[ A, P ]#" 2 01 00 00 0#[ Hi 0 ]Hermitenormal formexponentmatrixGeorge Labahn3Rational Invariants of Finite Abelian Groups19/37
Kernel? Use Hermite Normal FormDiagonal action : (α, β) Z6 Z3 :"4 1 1 34 3 6 00 3[ A, P ]exponentmatrix(Unimodular meansGeorge Labahn# 11011"100103002 1ViPiVnPn2201211100 #unimodularmultiplierW V 1 Z5 5 "32 01 00 00 0#[ Hi 0 ]Hermitenormal form)Rational Invariants of Finite Abelian Groups19/37
Hermite Normal FormDiagonal action : (α, β) Z6 Z3 :"4 1 1 34 3 60[ A, P ]exponentmatrix0 3# 1 1 0 1 110010"ViPi3 222 1 1 2VnPn#11100 unimodularmultiplier "3 2 01 00 00 0#[ Hi 0 ]Hermitenormal formNote : V not unique but can be normalized. Implies Vn is specialGeorge LabahnRational Invariants of Finite Abelian Groups20/37
Rational Invariants and Rewrite RulesTheoremA Zs n ,[A , P] · V [H, 0],"#"Vi VnWuV ,W V 1 Pi PnWdPuPd#(a) y [z1 , . . . , zn ]Vn form generating set of rational invariants.(b) V normalized : components of y [z1 , . . . , zn ]Vn are polynomials. 1(c) Rewrite rule : F K(z)A F(z) F(y(Wd Pd PuWu ))Why?George LabahnRational Invariants of Finite Abelian Groups21/37
Rational Invariants and Rewrite RulesTheoremA Zs n ,[A , P] · V [H, 0],"#"Vi VnWuV ,W V 1 Pi PnWdPuPd#(a) y [z1 , . . . , zn ]Vn form generating set of rational invariants.(b) V normalized : components of y [z1 , . . . , zn ]Vn are polynomials. 1(c) Rewrite rule : F K(z)A F(z) F(y(Wd Pd PuWhy?Wu ))vv Vn (Wd Pd P 1with v colspanZ Vn :u Wu )v. any term zzv 1 zVn (Wd Pd Pu Wu )v 1 (zVn )(Wd Pd Pu Wu )v 1 (y(Wd Pd PuWu ) v)Then use Lemma.George LabahnRational Invariants of Finite Abelian Groups21/37
Example : Rational Invariants for ZnG be cyclic group of permutations (1, 2, · · · , n).h iDiagonalizing matrix R(η) ηij , (η n-root of unity)hih i23.n 10 and P n .D:A 1 V 100.0n00.n ···.01000. 1 0 0 0 and W . . . 0 . . . 1 002 1103 101······00n 1 1···.0000.1001. n100.0 . Generating invariants are n 3g zVn zn1 , zn 21 z2 , z1 z3 , . . . , z1 zn 1 , zn ,Associated rewrite rules are 1 1z gWd Pd Pu Wu g1n , George Labahn g, . . . n 1 , gn ,n 21 nng1g1g2that is,Rational Invariants of Finite Abelian Groupszk gkn kg1 n22/37
Example : Rational Invariants for Zn ZnD:A 2 1 0 0 . . . . . . . . 0 0"11 1100.001213n0001n 1.0n001n 210#"n1and P 00·········2n 301.111212······.······200.#0. V and W then:nn 210.1012111 & 11000.012000.0············0130 111n 10 1···10 100.11. n02 10.00 n 110.0Generating invariants are 2 n 3n 3 2n 2g zVn zn1 , zn2 , z1 zn 22 z3 , z1 z2 z4 , . . . , z1 z2 zn 1 , z1 z2 zn ,Associated rewrite rules are 1 1 1z gWd Pd Pu Wu g1n , g2n , George Labahn ., .,,1 n 2n 3 2n 2 1 nnnnnng1 g2g1 g2g1 g2g3gn 1Rational Invariants of Finite Abelian Groupsgn23/37
Solving Polynomial SystemsUsing invariants and rewrite rules(A P)-degree and (A P)-homogeneousSolving invariant systemsGeorge LabahnRational Invariants of Finite Abelian Groups24/37
Example : Invariant Dynamic SystemConsider system of polynomial equations (c parameter)11 cx1 x1 x22 x1 x32 01 cx2 x2 x12 x2 x32 01 cx3 x3 x12 x3 x22 01George LabahnSteady state for Neural network model [ Noonburg SIAM Num Anal 1989]Rational Invariants of Finite Abelian Groups25/37
Example : Invariant Dynamic SystemConsider system of polynomial equations (c parameter)11 cx1 x1 x22 x1 x32 01 cx2 x2 x12 x2 x32 01 cx3 x3 x12 x3 x22 0Solution space of system is invariant under the permutation(x1 , x2 , x3 ) (x2 , x3 , x1 ).However no polynomial is invariant under the permutation.1George LabahnSteady state for Neural network model [ Noonburg SIAM Num Anal 1989]Rational Invariants of Finite Abelian Groups25/37
(A, P)-homogeneity (Faugère and Svartz)Definition(i) deg(A,P) (zu ) A · u mod P(ii) f K[z, z 1 ] can be written asXfdf d Zterms in fd deg(A,P) d ( homogeneous of (A, P)-degree d)Lemmaf K[z, z 1 ] is (A, P)-homogeneous of (A, P)-degree d iff f λA z λd f (z)for all λ U.George LabahnRational Invariants of Finite Abelian Groups26/37
Solving via (A, P)-homogenous componentsTheoremLet F K[z, z 1 ] and F h {fd f F, d Zp1 . . . Zps } set ofhomogeneous components of F.If set of toric zeros of F is invariant by the diagonal action of Udefined by A then it is equal to toric zeros of F h .George LabahnRational Invariants of Finite Abelian Groups27/37
Solving via (A, P)-homogenous componentsTheoremLet F K[z, z 1 ] and F h {fd f F, d Zp1 . . . Zps } set ofhomogeneous components of F.If set of toric zeros of F is invariant by the diagonal action of Udefined by A then it is equal to toric zeros of F h .Why:Since f (λA z) Pdd λ fd (z) for all λ U we have a square linear system f (λA z)λ U λdλ U,d Z fd d Z . With an appropriate ordering of the elements of U and Z the square matrix λdis the Kronecker product ofλ U,d Z (k 1)(l 1), for 1 i s and ξi a primitive pi th root of unity. So it is invertible.the Vandermonde matrices ξi1 k,l piGeorge LabahnRational Invariants of Finite Abelian Groups27/37
Example : Neural NetworkRecall Neural Network system (c is a parameter):1 cx1 x1 x22 x1 x32 01 cx2 x2 x12 x2 x32 01 cx3 x3 x12 x3 x22(1) 0(i) Zeros invariant under permutation σ (321).(ii) Diagonal action : exponents A [1 2 0]; order P [3].George LabahnRational Invariants of Finite Abelian Groups28/37
Example : Neural Network(iii) Change coordinates via x R · z gives0 f0 f0 ξ f1 ξ2 f20 f1 f0 ξ2 f1 ξ f20 f2 f0 f1 f2wheref0 1 cz3 z31 z32 2z33f1 cz1 3z21 z2 3z22 z3f2 cz2 3z1 z22 3z21 z3 .(iv) Each fi is (A, P)-homogeneous of degree i, for 0 i 2.George LabahnRational Invariants of Finite Abelian Groups29/37
Example (cont.)(i) What about non-toric zeros? Localize at z1 (ii) The reduced system corresponding to f0 ,0 1 y1 cy3 2y23 y32y1,0 c 3y2 3f1 f2z1 , z21y22 y3,y1 is0 3y3 cy2 y22 3 .y1 y1(iii) This system has 6 2 4 zeros : union of triangular setsy3 0,y2 c,3c3y21 y1 0;27and162 c y43 54 y33 81 c2 y23 108 c y3 4 c3 27 0,y2 81 c14 c393 c2c (70 c3 243)y3 y2 y3 49 c3 27 3 49c3 27 3 2(49 c3 27)6 (49 c3 27)1cy1 y33 y3 .22George LabahnRational Invariants of Finite Abelian Groups30/37
Example (cont.)Original system has 6 orbits of zeros, that is 18 solutions, wherez1 ξ2 x1 ξx2 x3 , 0.(i) Given sol. (y1 , y2 , y3 ) orbit : by solve triangular system:z31 y1 , z1 z2 y2 , z3 y3 .(ii) With x R z get 18 solutions of the system with 6 orbits.What about z1 0?(iii) Here, there are three solutions satisfyingz1 0,z2 0,2 z33 c z3 1 0.(iv) They each form an orbit. The corresponding solutions :x1 x2 x3 η, for 2 η3 c η 1 0.Total number of solutions : 21.George LabahnRational Invariants of Finite Abelian Groups31/37
Determining groups of homogeneityMatrix of exponents of polynomial systemSmith Form and finding A and PGeorge LabahnRational Invariants of Finite Abelian Groups32/37
Matrix of Exponents of SystemRational functions in K(z1 , . . . , zn ).K matrix of exponents.f1 z 1 2 z 2 2 z 3 2 z 2 3 z 1 z 2 z 3 8f2 z 1 2 z 2 2 z 3 2 z 2 3 7f3 z61 z32 z33 3z41 z42 z3 z61 z32 32z31Matrix of Exponents : 2 0 1 2 0 3 1 3 3 K 2 3 1 2 3 3 4 0 3 . 2 0 1 2 0 3 1 0 0George LabahnRational Invariants of Finite Abelian Groups33/37
Smith Normal Form and finding A and PhiSmith normal form K : U · K · V diag (1, ., 1, p1 , ., ps ) 0TheoremPartition: U "CA#andhiU 1 U0 U1(i) F invariants for diagonal actionP diag (p1 , . . . , ps ),hA the last s rows of U.i(ii) [y1 , . . . , yn ] z U0 U1 P minimal generating invariants(iii) Rewrite rule : for any invariant f K(z) of (A, P): !CP 1 A.f (z) f yGeorge LabahnRational Invariants of Finite Abelian Groups34/37
Matrix from previous example: 2 0 1 2 0 3 1 3 3 3K 2 3 1 2 3 3 4 0 2 0 1 2 0 3 1 00 .Smith normal form: 0 0 1 0 0 0 0 0 0 0 0 1 1 1 · K · V 0 3 0 0 0 0 0 0 0 2 0 0 3 0 0 0 0 0 01 1 0The underlying symmetry group is Z3 Z3 .##""3 01 1 1and P A 0 31 2 0Invariant exponents: 0 1 0 Vn [ U0 U1 P ] 1 0 3 1 33George LabahnRational Invariants of Finite Abelian Groups35/37
Matrix from previous example: 2 0 1 2 0 3 1 3 3 K 2 3 1 2 3 3 4 0 3 . 2 0 1 2 0 3 1 0 0Smith normal form: 0 0 1 0 0 0 0 0 0 0 0 1 1 1 · K · V 30 0 0 0 0 0 2 1 1 03 0 0 0 0 0 0The underlying symmetry group is Z3 Z3 .#""1 1 13and P A 1 2 00Invariant exponents: 1 0 0 Vn [ U0 U1 P ] 1 0 3 1 3 3George Labahn03# 3 1 0 0 1 0 0 0 1Rational Invariants of Finite Abelian Groups 36/37
Future Research Directions(i) Extend to parameterized and dynamic systems(ii) Extend from Finite Abelian to Finite Solvable Group actions- e.g. Neural network example invariant under S3 .(iii) Combine scaling symmetries with finite diagonal actions- makes use of Smith Normal FormGeorge LabahnRational Invariants of Finite Abelian Groups37/37
Rational Invariants of Finite Abelian Groups (1) Given finite, abelian group G GL(n,K) acting on Kn - construct rational invariants of action rational invariant : f K(x) : f( g ·x ) f( x ) g G - determine rewrite rules for this action (2) Given system of polynomial equations - if have group action then ‘reduce .
invariants which generate the ring of invariants of genus one models over a eld. Fisher considered these invariants over the eld of rational numbers and normalized them such that they are moreover de ned over the integers. We provide an alternative way to express these normalized invariants using a natural connection to modular forms. In
invariants of Xthat are most relevant for arithmetic investigations. There are two natural types of invariants: birational invariants, i.e., invariants of the function field F(X), and projective geometry invariants, i.e., those arising . is the multiplicative group of rational functions of X, .
Rational Invariants of Algebraic Group Actions Evelyne Hubert INRIA M editerran ee TEDI 5 - Marseille 2016 Partly based on joint works with Irina Kogan or George Labahn Contents 1 Computing rational invariants 2 . Rational invariants of scalings from Hermite normal forms. In
Rational Invariants of Meta-abelian Groups of Linear Automorphisms* MOWAFFAQ HAJJA Yarmouk University, Irbid, Jordan Communicated by R. G. Swan Received April 6, 1981 INTRODUCTION Let k be an algebraically closed field of characteristic zero, G a finite group and V a finite-dimensional kG-module.
Rational and Algebraic Invariants of a Group Action Evelyne Hubert Kolchin Seminar in Differential Algebra. December 2006 Synopsis Propose algebraic constructions of a generating set of invariants that comes with a simple rewriting algorithm. E. Hubert and I. Kogan, Rational Invariants of a Group Action. Construction and Rewriting.
generating rational invariants. The second construction is given after the introduction of the cross-section to the orbits in Section 3.Section 4 pro vides additional examples. 2. Graph of a gr oup action and rational invariants W e give a deÞnition of a rational action of an algebraic group on an afÞne space. T w o
Rational Rational Rational Irrational Irrational Rational 13. 2 13 14. 0.42̅̅̅̅ 15. 0.39 16. 100 17. 16 18. 43 Rational Rational Rational Rational Rational Irrational 19. If the number 0.77 is displayed on a calculator that can only display ten digits, do we know whether it is rational or irrational?
1 Advanced Engineering Mathematics C. Ray Wylie, Louis C. Barrett McGraw-Hill Book Co 6th Edition, 1995 2 Introductory Methods of Numerical Analysis S. S. Sastry Prentice Hall of India 4th Edition 2010 3 Higher Engineering Mathematics B.V. Ramana McGraw-Hill 11 th Edition,2010 4 A Text Book of Engineering Mathematics N. P. Bali and Manish Goyal Laxmi Publications 2014 5 Advanced Engineering .
