The original expression can be written as, 4-2 Greatest common divisor of univariate polynomials. Let us consider the following two polynomials.

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1 Algebraic Geodesy and Geoinformatics 2009 PART I - METHODS 4 Groebner Basis 4- Greatest common divisor of integers Let us consider the following integer numbers 2, 20 and 8. Factorize them, Clear@"Global *"D 882, 2<, 83, << or CenterDot HSuperscript %L 22 3 similarly 882, 2<, 85, << CenterDot HSuperscript %L 22 5 and 882, <, 83, 2<< CenterDot HSuperscript %L 2 32 From there the common factors give us the greatest common divisors, 2 * 30 * 50 2 One can compute it directly GCD@2, 20, 8D 2 The original expression can be written as,

2 2 Groebner_Basis_04.nb The original expression can be written as, 2, 20, 8 6, 0, Greatest common divisor of univariate polynomials Let us consider the following two polynomials. f 2 x 2 x 4 ; f2 3 2 x x 2 ; If we are looking for the the common roots, then we should find out the greatest common divisor of the polynomials. This is the highest degree polynomial that divides the two polynomials exactly. The roots of this polynomial will be the common roots! We employ elimination technique, which is essentially analog for polynomials of the Euclidean algorithm for integers. Let f f; f2 f2; Step. f > f2, since Exponent f, x Exponent f2, x Let us multiply f2 by x 2 and add to f in order to eliminate the highest order term, x 4 from f. With other words we reduce the highest order exponent of f, f f x 2 f2 Expand x 2 2 x 3 Step 2. f > f2 Similarly f f 2 x f2 Expand 6 x 5 x 2 Now, both polynomials are second order, although f f2, f 5 Expand 5 6 x 5 Step 3. f > f2 x2 we need to eliminate x 2 from f f f 5 f2 Expand 6 6 x Step 4. f < f2 now we eliminate x 2 from f2

3 Groebner_Basis_04.nb f2 x f 6 f2 Expand 3 3 x But now it is clear that, f 6 f2 3 Expand and the greatest common divisor is, f 6 Expand x Their greatest common divisors with the built in function, gcd PolynomialGCD f, f2 x and the reduced polynomials, fr PolynomialQuotient f, x, x x x 2 x 3 f2r PolynomialQuotient f2, x, x 3 x Consequently the original polynomials can be expressed as f, f2 x fr, f2r Simplify and Solve gcd 0, x Flatten x Therefore the common root of the two polynomial is x =. 4-3 Lexicographic order In order to find the greatest common divisor of multivariate polynomials we may use the similar elimination technique but now, the problem is the ordering. Which polynomial is greater than the other? In Mathematica we use lexicographic order, x > y, i.e., x comes before y. Let p = x and p 2 = y 5 z 9 two polynomials, then p is greater than p 2. Considering the following polynomial poly 2 x 2 y 8 3 x 5 y z 4 x y z 3 x y 4 ; The list of the monomials in lexicographic order, MonomialList poly, x, y, z 3 x 5 y z 4, 2 x 2 y 8, x 2 y 5, x y z 3 This is the list of the monomials represented as exponent vectors and coefficients,

4 4 Groebner_Basis_04.nb CoefficientRules poly, x, y, z 5,, 4 3, 2, 8, 0 2, 2, 5, 0,,, 3 or CoefficientRules poly 5,, 4 3, 2, 8, 0 2, 2, 5, 0,,, 3 By default the monomials are sorted lexicographically and given in the decreasing order. In this example, 5,, 4 corresponding to x 5 y x 4 is taken to precede 2, 8, 0 corresponding to x 2 y 8 z 0 by the second element. The numbers - 3 and 2 are the corresponding coefficients of these terms. The following functions can be defined: - the leading coefficient, LC p_ : CoefficientRules p, 2 LC poly 3 - the leading term, LT p_ : MonomialList p LT poly 3 x 5 y z 4 - the leading monomial, LM p_ : LT p LC p Simplify LM poly x 5 y z Greatest common divisor of multivariate polynomials In that case the normal Euclidean algorithm does not work. Let g x y; g2 x y; then PolynomialGCD g, g2 It would mean, that g and g2 are relativ primes. However they are not, G, G2 GroebnerBasis g, g2, x, y y 2, x y

5 Groebner_Basis_04.nb where G and G2 represent the Groebner basis. The polynomials g(x, y) and g2(x, y) can be expressed as the linear combination of G and G2. and g, y. G, G2 x y y 2 y x y Simplify g2 0,. G, G2 Formally we can consider the Groebner basis as the greatest common divisors of g and g2, g, g2, y, 0,. G, G2 Simplify 4-5 Employing Groebner basis for solving polynomial system The Grobner basis of the polynomial system, {G(x,y), G2(x,y)} can be considered as another but equivalent representation of {g (x, y), g2 (x, y)}, in sense of the roots of the system, since g and g2 can be expressed as linear combination of the members of the Groebner basis. Consider F (x, y) = {f, f2}, where f x y 2 y; f2 2 y 2 x 2 ; Now we seek a simplified representation, G(x, y), where one of the polynomials is an univariate one, see Section ! G GroebnerBasis f, f2, y, x 2 x 2 x 3, 2 y x y, x 2 2 y 2 then G() can be solved numerically, Roots G 0, x x 2 x 0 x 0 and substituting them in the third element of the set G to solve for y, and Roots G 3. x 2 0, y y 2 y 2 Roots G 3. x 0 0, y y 0 y 0 Solving the system directly, we get the same pairs of roots,

6 6 Groebner_Basis_04.nb Solve f 0, f2 0, x, y x 0, y 0, x 0, y 0, y 2, x 2, y 2, x Ideal of polynomials The systems of G(x, y) and F(x,y) can be considered as two different bases or generators of the same ideal I. We learnt that polynomials are elements of a ring and they satisfy the ring axioms of addition and substruction. The computation of the Groebner basis is achieved by the capability to manipulate the polynomials to generate ideals. It is easy to test whether a polynomial belongs to the ideal generated by a set of polynomials. If f is a polynomial and p i is a set of polynomials generating an ideal, then f can be expressed as where r is the minimal remainder modulo. If this remainder modulo is zero, then f belongs to the ideal. Now let us check whether f and f2 belong to the ideal generated by the Groebner basis, G. and c, r PolynomialReduce f, G, x, y 0,, 0, 0 f c.g r r 0 Similarly, Vice versa, and c, r PolynomialReduce f2, G, x, y 0, 0,, 0 PolynomialReduce G, f, f2, x, y 2 y, 2 x, 0 PolynomialReduce G 2, f, f2, x, y, 0, 0 It means that the elements of G belong to the ideal F = {f, f2}. Consequently F and G generate the same ideal. 4-7 Buchberger algorithm The algorithm for computing Groebner basis uses S polynomial function. Let consider two polynomials, f and g, then Clear S S f_, g_ : PolynomialLCM LM f, LM g f LT f g LT g Simplify

7 Groebner_Basis_04.nb For example, let p x 2 y; p2 x y; This S polynomial function will reduce p to p3, employing p2. p3 S p, p2 y x y Now, on one hand where p3 p x p2 Simplify therefore p 3 belongs to the same ideal, which generated by p and p 2. On the other hand p p3 x p2 Simplify which means that the ideal I p, p 2 is the same as I p 2, p 3, but in different representation. The new basis is, p 2, p 3 = x + y, y - x y Because, the leading monomials are LM(p 2 ) = x and LM( p 3 ) = x y, further reduction is possible, LM p2 x LM p3 x y p4 S p2, p3 Expand y y 2 The new basis is, Now, because LM p2 x and LM p4 y 2

8 8 Groebner_Basis_04.nb no further reduction is possible. Using built-in function GroebnerBasis p, p2, x, y y y 2, x y But of course GroebnerBasis p2, p3, x, y y y 2, x y is also true. 4-8 Mathematica computation of Groebner basis In Mathematica Groebner basis can be easily computed, eqs x 2 3 x 4 y, 2 y 2 4 x 5 y 9 ; GroebnerBasis eqs, x, y y 37 y 2 20 y 3 4 y 4, 9 4 x 5 y 2 y 2 By Hilbert s Nullstellensatz, if the ideal is {} then the polynomials have no common zero, GroebnerBasis x y, x ^ 2, y ^ 2 2 x, x, y The default options of this function are, Options GroebnerBasis CoefficientDomain Automatic, Method Automatic, Modulus 0, MonomialOrder Lexicographic, ParameterVariables, Sort False, Tolerance 0 eqs = {2 x^4 y + y^3 x^3 - x z^2 +, x^2 + y^2 z^3 -, x^2 y - 7 y^3 z^2 + y^2 z^3}; The default method is the Groebner - Walk, which sometimes works faster than the Buchberger algorithm (gb = GroebnerBasis[eqs, {x, y, z}]); // Timing 0.203, Null (gb = GroebnerBasis[eqs, {x, y, z},method-> "Buchberger"]); // Timing 0.235, Null The basis consists of 3 polynomials, where the exponents of the variables x, y and z respectively Exponent, x, y, z & gb 0, 0, 44, 0,, 43,, 0, 43 The first polynomial in the basis is an univariate polynomial of degree 44,

9 Groebner_Basis_04.nb gb z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z 40 2 z 4 96 z z 43 4 z 44 Changing the coeffient domain from infinite precision to number with finite precision, the computation time can be reduced, gbn GroebnerBasis N eqs, 300, x, y, z, CoefficientDomain InexactNumbers ; Timing 0.87, Null The result is practically the same and NSolve gb 0, z z 7.865, z , z , z , z , z , z.96926, z , z , z.24096, z., z , z , z , z , z , z , z , z , z , z , z , z 0.., z 0.., z , z , z , z , z , z , z , z , z , z , z , z , z , z., z.285, z , z , z , z , z NSolve gbn 0, z, 6 z , z , z , z , z , z , z.96926, z , z , z.24096, z.00000, z , z , z , z , z , z , z , z , z , z , z , z.00000, z.00000, z , z , z , z , z , z , z , z , z , z , z , z , z , z.00000, z.28500, z , z , z , z , z However, we need to use high precision, (gbn = GroebnerBasis[N[eqs, 200], {x, y, z}, CoefficientDomain -> InexactNumbers]); // Timing GroebnerBasis::fltgb: Excessive loss of precision during computation. 0.72, Null

10 0 Groebner_Basis_04.nb The order of the variables has influence on the form of the basis, or GroebnerBasis x y z, x 2 y z ^ 3, x ^ 2 2 y ^ 3 z, x, y, z 27 z 2 z 2 2 z 3 2 z 4 6 z 5 3 z 6 6 z 7 2 z 9, 3 y z z 3, 3 x 2 z z 3 GroebnerBasis x y z, x 2 y z ^ 3, x ^ 2 2 y ^ 3 z, z, y, x 27 x 8 x x x 4 86 x x 6 8 x 7 2 x 8 8 x 9, x x x x x x x x y, x x x x x x x x z Computation time may be reduced by using reverse lexicographic order, Timing gblex GroebnerBasis x ^ 4 y z, x 2 y ^ 3 5 z ^ 3, x ^ 2 7 y ^ 3 z ^ 4, z, y, x, MonomialOrder Lexicographic ; 0.094, Null Timing gbdegrevlex GroebnerBasis x ^ 4 y z, x 2 y ^ 3 5 z ^ 3, x ^ 2 7 y ^ 3 z ^ 4, z, y, x, MonomialOrder DegreeReverseLexicographic ; 0., Null Both basis have 4 polynomials, Map[Length, {gblex,gbdegrevlex}] 4, 4 The order of the polynomials are considerably lower in the second case, Map[Length, {gblex, gbdegrevlex}, {2}] 28, 46, 46, 3, 3, 3, 4, 8 However, in case of the reverse lexicographic order our system is not triangular, (Exponent[#, {x, y, z}]& /@ #)& /@ {gblex, gbdegrevlex} 46, 0, 0, 45,, 0, 45, 3, 0, 4,,,, 3, 3, 4,,, 2, 3,, 2, 6, 2 gbdegrevlex x 2 y 3 5 z 3, x 4 y z, 5 x 2 35 y 3 x z 2 y 3 z, x y 3 6 x y 3 6 y x z 750 x 2 z 350 x z 2 00 x 2 z 2 Let us set ordering option to elimination order for an interesting example: What is the largest area that a hexagon of unit diameter can have? Interestingly, the largest area is not the one of a regular hexagon. It can be shown that the hexagon we are looking for must have mirror symmetry and be of the following form, see Fig.5.. Without loss of generality we can use the following parametrization of the hexagon, Clear["x*", "y*"]; p = {0, 0}; p2 = { x2, y2}; p3 = { x3, y3}; p4 = {0, }; p5 = {-x3, y3}; p6 = {-x2, y2};

11 Groebner_Basis_04.nb Block[{x2 = 0.5, y2 = 0.402, x3 = 0.343, y3 = 0.939}, Show[Graphics[{{Hue[0], Thickness[0.0], (* outline *) Line[{p, p2, p3, p4, p5, p6, p}]}, {GrayLevel[0], Thickness[0.002], (* diagonals *) Line[{p, p3}], Line[{p, p5}], Line[{p2, p5}], Line[{p3, p6}], Line[{p, p4}], Line[{p2, p5}]}}], PlotRange -> All, Frame ->, AspectRatio -> Automatic]] Fig. 5. Optimal hexagon of unit diameter The above hexagon has four degrees freedom, the coordinates of the points p2 and p3. It follows from elementary geometry that the area of the hexagon is given by x 3 x 3 y 2 x 2 y 3. We take the unit diameter conditions into account using Λ, Λ 2, and Λ 3 Lagrange multipliers, area = x3 - x3 y2 + x2 y3; L = area + Λ (#.#&[p - p3] - ) + Λ2 (#.#&[p2 - p5] - ) + Λ3 (#.#&[p2 - p6] - ) x3 x3 y2 x2 y3 x3 2 y3 2 Λ x2 x3 2 y2 y3 2 Λ2 4 x2 2 Λ3 The maximum values of the area is represented with the variable area. In order to get this value, we eliminate all of the other variables from the Groebner basis, we call it as reduced Groebner basis, see also in Section The variables {area,x2, y2, x3, y3, Λ, Λ 2, Λ 3 } will be eliminated and in addition the option for elimination order is MonomialOrder EliminationOrder, GroebnerBasis[ {area - area, D[L, x2], D[L, y2], D[L, x3], D[L, y3], D[L, Λ], D[L, Λ2], D[L, Λ3]}, {area}, {x2, y2, x3, y3, Λ, Λ2, Λ3}, MonomialOrder -> EliminationOrder] // Factor area area area area area area area area area area area area area area area area area area area area 0 Remark: In this case, the variables to be eliminated are represented by the third list in the function. For example, computing the reduced Groebner basis for the variable area, the variables {x2, y2, x3, y3, Λ, Λ 2, Λ 3 } should be eliminated. The root which is real and larger than the area of a regular hexagon,

12 2 Groebner_Basis_04.nb Select[NSolve[%[[]] == 0, area], Im[(area /. #)] == 0 && (* larger than regular hexagon *) 3 Sqrt[3]/8 < (area /. #)&][[]] area The area of the regular hexagon is, 3 Sqrt It is nearly 4 % smaller the the area of our hexagon above. Here we present a function, which is useful if one wants to study the effect of - the order of variables - the type of elimination order - the type of the coeffient domain and other options of the GroebnerBasis function implemented in Mathematica GroebnerBasisStatistics polys_, vars_, opts : Module time, gb, numberpolys, numberterms, totaldegrees, w, maxcoeffs, time First Timing gb GroebnerBasis polys, vars, opts ; numberpolys Length gb ; numberterms Length gb; totaldegrees Exponent, w & gb. Thread vars Table Random, Length vars w ; maxcoeffs Max Abs List. Thread Variables polys & gb; TableForm " ", " ", " ", " ", " ", " ", time, numberpolys "polys", " ", " ", " ", " ", "terms", "total", "max", "in poly", "degrees", "coeffs", " ", " ", " ", numberterms, totaldegrees, maxcoeffs Let us consider the following polynomial system, polys x 6 y 4 z 3, x 5 y 3 z 2 ; GroebnerBasisStatistics polys, y, z, x polys terms total max in poly degrees coeffs Let us change the order of the variables,

13 Groebner_Basis_04.nb GroebnerBasisStatistics polys, z, y, x polys terms total max in poly degrees coeffs This result is better. Let us give another variable order, GroebnerBasisStatistics polys, x, y, z polys terms total max in poly degrees coeffs Now we got the best result. Let us change the monomial order, GroebnerBasisStatistics polys, x, y, z, MonomialOrder DegreeReverseLexicographic 0. 3 polys terms total max in poly degrees coeffs The result could be improved. Let us consider the variable z as constant and change the coefficient domain, GroebnerBasisStatistics polys, x, y, CoefficientDomain RationalFunctions polys terms total max in poly degrees coeffs Now the basis consists of only two polynomials.

14 4 Groebner_Basis_04.nb 4-9 Examples D Resection Let us solve the Grunert s distance equations (see the text book), p x 2 x x2 x2 2 d 0; p2 x2 2 x2 x3 x3 2 d 0; p3 x3 2 x x3 x 2 d 0; The Groebner basis Gb GroebnerBasis p, p2, p3, x, x2, x3, ParameterVariables d ; TableForm Gb d x3 x3 3 d x2 2 x2 x3 x3 2 d x d x2 x x3 2 x2 x3 2 d x x2 x x3 x2 x3 2 x3 2 d x 2 x x3 x3 2 This is a triangular system. Let us solve it, Solve Map 0 &, Gb, x, x2, x3 x 0, x2 d, x3 d, x 0, x2 d, x3 d, x d, x2 0, x3 d, x d, x2 d, x3 0, x d, x2 d, x3 d, x d, x2 d, x3 d, x d, x2 0, x3 d, x d, x2 d, x3 0, x d, x2 d, x3 d, x d, x2 d, x3 d From geometric point of view, only the positive solutions should be considered, Last x d, x2 d, x3 d The full symmetry of the problem can be revealed, when reduced Groebner Basis is computed, Gbx GroebnerBasis p, p2, p3, x, x2, x3, x2, x3 d x x 3 Gbx2 GroebnerBasis p, p2, p3, x, x2, x3, x, x3 d x2 x2 3 Gbx3 GroebnerBasis p, p2, p3, x, x2, x3, x, x2 d x3 x3 3 In general Clear X

15 Groebner_Basis_04.nb Solve d X X 3 0, X X 0, X d, X d D Helmert transformation Transformation from one system of coordinates to another is a very useful operation that is used frequently in photogrammetry, geodesy, and surveying. Considering two dimensional space, the transformation from one cartesian coordinate system to another with rotation and scale or Let us suppose, that we have measurements for three corresponding data pairs (x i, y i ) (X i, Y i ), i =,2,3, respectively, see Table 5. Table 5. Measured data pairs in the two systems We require the least square estimates of the transformation parameters, Α and Β and simultaneously the adjustment of the coordinates, namely x a, x b, x c,x a, X b,and X c. Applying the two equations of the transformation for each of the three point - pairs with the adjusted values, we get, fa Α xa dxa Β ya Xa dxa ; ga Β xa dxa Α ya Ya; fb Α xb dxb Β yb Xb dxb ; gb Β xb dxb Α yb Yb; fc Α xc dxc Β yc Xc dxc ; gc Β xc dxc Α yc Yc; In these 6 equations there are 8 unknowns, the adjustments (dx a, dx b, dx c,dx a, dx b, dx c ) and the two parameters (Α, Β) to be estimated. This underdetermined system can be transformed into a constrained minimization problem formulated with Lagrange multipliers, Λ,..., Λ 6 F dxa 2 dxb 2 dxc 2 dxa 2 dxb 2 dxc 2 Λ, Λ2, Λ3, Λ4, Λ5, Λ6. fa, ga, fb, gb, fc, gc ; Now, we have 4 unknowns, d dxa, dxb, dxc, dxa, dxb, dxc, Α, Β, Λ, Λ2, Λ3, Λ4, Λ5, Λ6 ; The necessary condition for the existence of the optimum provides the following polynomials,

16 6 Groebner_Basis_04.nb eq Table D F, d i, i,, 4 2 dxa Α Λ Β Λ2, 2 dxb Α Λ3 Β Λ4, 2 dxc Α Λ5 Β Λ6, 2 dxa Λ, 2 dxb Λ3, 2 dxc Λ5, dxa xa Λ ya Λ2 dxb xb Λ3 yb Λ4 dxc xc Λ5 yc Λ6, ya Λ dxa xa Λ2 yb Λ3 dxb xb Λ4 yc Λ5 dxc xc Λ6, dxa Xa dxa xa Α ya Β, Ya ya Α dxa xa Β, dxb Xb dxb xb Α yb Β, Yb yb Α dxb xb Β, dxc Xc dxc xc Α yc Β, Yc yc Α dxc xc Β Using rational data providing infinite precision, then xa 0; ya ; Xa 2 0; Ya 0; xb ; yb 0; Xb ; Yb 2; xc ; yc ; Xc 9 0; Yc 28 0; eq 2 dxa Α Λ Β Λ2, 2 dxb Α Λ3 Β Λ4, 2 dxc Α Λ5 Β Λ6, 2 dxa Λ, 2 dxb Λ3, 2 dxc Λ5, dxa Λ Λ2 dxb Λ3 dxc Λ5 Λ6, Λ dxa Λ2 dxb Λ4 Λ5 dxc Λ6, 2 dxa dxa Α Β, Α dxa Β, 0 0 dxb dxb Α, 2 dxb Β, 9 Let us compute the Groebner basis, gr GroebnerBasis eq, d ; Timing 0.203, Null This basis consists of 4 polynomials, Length gr 4 and they represent a triangular system, Table Exponent gr i, d, i,, 4 0 dxc dxc Α Β, 4 5 Α dxc Β 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,, 0, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0,, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 0,, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0,, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0,, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0,, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0,, 0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0,, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0,, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 0,, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5,, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5 The first basis is a monomial of the variable Λ 6, namely gr Λ Λ Λ Λ Λ Λ6 6 This is a polynomial with degree of six for Λ 6. The solutions are

17 Groebner_Basis_04.nb solλ6 NSolve gr 0, Λ6 Λ , Λ , Λ , Λ , Λ , Λ Let us consider the positive, real solution, Λ6 Λ6. solλ Then the values of the other variables can be easily computed via successive elimination. d dxa, dxb, dxc, dxa, dxb, dxc, Α, Β, Λ, Λ2, Λ3, Λ4, Λ5, We are interested in the first 8 variables, or with Table d i. Solve Reverse gr i 0, d i, i,, 8 Flatten , , , , , ,.0574, dr Take d,, 8 dxa, dxb, dxc, dxa, dxb, dxc, Α, Β MapThread 2 &, dr, dxa , dxb , dxc , dxa , dxb , dxc , Α.0574, Β In order to check our result, we solve this constrained optimization problem by direct global minimization with the built in function NMinimize of Mathematica, too. NMinimize dxa 2 dxb 2 dxc 2 dxa 2 dxb 2 dxc 2, fa 0, fb 0, fc 0, ga 0, gb 0, gc 0, dr Timing 0.328, , dxa , dxb , dxc , dxa , dxb , dxc , Α.0574, Β So we have got the same result, but the computation time is longer than in case of Groebner basis solution.

Groebner Bases, Toric Ideals and Integer Programming: An Application to Economics. Tan Tran Junior Major-Economics& Mathematics

Groebner Bases, Toric Ideals and Integer Programming: An Application to Economics Tan Tran Junior Major-Economics& Mathematics History Groebner bases were developed by Buchberger in 1965, who later named