1. Narapong Srivisal, Swarthmore College Class of 2007 Division Algorithm Fix a monomial order > in k[x 1, …, x n ]. Let F = (f 1, …, f s ) be an ordered s-tuple of polynomials in k[x 1, …, x n ]. Then every can be written as f = h 1 f 1 + … + h s f s + where h i,.. For all i, h i f i = 0 or, and is equal to zero or a linear combination of monomials, non of which is divisible by LT > (f i ),…, LT > (f s ). Call a remainder of f on division by F Gröbner Basis Let be an ideal. A Gröbner basis for I is a finite collection of polynomials s.t., LT(f) is divisible by LT(g i ) for some i. Theorem (Uniqueness of Remainders) A division of by a Gröbner basis for I produces an expression f = g + r, where g I, and no term in the remainder r is divisible by leading terms of any elements of I. If f = g’ + r’ is any other such expression, then r = r’. Theorem A Gröbner basis for I is a basis for I. Theorem (Buchberger’s Criterion) is a Gröbner basis for iff,. Definition S-Polynomial of, denoted S(f,g), is the polynomial where is the least common multiple monomial of LT(f) and LT(g). A monomial in x 1,x 2,…,x n is a product of the form,where each is a non-negative integer. The total degree of this monomial is equal to. Let donote,where A Polynomial with coefficients in a field k is a finite linear combination of monomials. Write k[x 1, …, x n ] for the set of all polynomials in x 1, x 2, …, x n with coefficients in k. Definition Lexicographic Order: Fix variable order x 1 > x 2 > … > x n. Let. Then, if the leftmost entry of is positive. The leading term of with respect to a monomial order >, LT > (f), is,where is the largest monomial appearing in f in the ordering >, and x α is the leading monomial. Definition Affine Variety V(f 1,…, f s ) Theorem If, then V(f 1,…, f s ) = V(g 1,…, g t ). Definition Let. The m th elimination ideal I m is the ideal of defined by Theorem I m is an ideal for of k[x m+1, x m+2, …, x n ] The Extension Theorem Let, where k is an algebraically closed field Let I 1 be the first elimination ideal of I. Write where N i ≥ 0 and is non-zero. Suppose (a 2, a 3, …, a n ). If (a 2, a 3, …, a n ), then References Cox, David A. and John Little and Donal O’Shea. (2004) Using Algebraic Geometry (2 nd Edt). Springer. Cox, David A. and John Little and Donal O’Shea. (1992) Ideals, Varieties, and Algorithm. Springer-Verlag, New York. http://www.scch.at The Elimination Theorem Let G be a Gröbner basis for I with respect to lexicographic order, where x 1 > x 2 > … > x n. Then, for every 0 ≤ m ≤ n, the set is a Gröbner basis for the m th elimination ideal I m ProofLet G = {g 1,..., g t }. Fix 0 ≤ m ≤ n. Suppose G m = {g 1,..., g r }, where r < t i). WTS G m is a basis for I m + Since and I m is an ideal, then + Pick any. By Division Algorithm and theorem, f = h 1 g 1 + h 2 g 2 + … + h r g r + h (r + 1) g (r+1) + … + h t g t However, for each i > r, g i has at least one term involving x 1, x 2, …, or x m. Thus,,. Therefore,, by Division Algorithm. ii).WTS G m is a Gröbner basis for I m Let 1≤ i< j≤ r. Since, so are the leading terms and the l.c.m. monomail of the leading terms,. Then, Therefore,, and G m is a Gröbner basis for I m by theorem. By i and ii, the theorem has been proved. Let f 1 = f 2 = f 3 = … = f s = 0 be the system of polynomial equations in that we would like to solve eg. in,0 = x 2 + y 2 + z 2 – 4= f 1 0 = x 2 + 2y 2 – 5= f 2 0 = xz – 1= f 3 The set of all solutions of f 1 = f 2 = f 3 = … = f s = 0 is V(f 1, f 2, f 3, …, f s ) Change the basis for I to a Gröbner basis, G = {g 1, g 2,g 3,..., g t }, with respect to lexicographic order by using the Buchberger‘s Criterion. Below is an example of a simple programing to find a Gröbner basis By the Elimination Theorem, G m = is a Gröbner basis for the m th elimination ideal I m. In other words, a lex Gröbner basis G successively eliminate variables. eg. A lex Gröbner basis for the example system (fix monomial order x > y > z) is Find partial solutions of the system equation, V(I m ), and extend to V(I m-1 ), starting with m = n – 1 until receiving the complete solutions, V(g 1, g 2,..., g t ). eg. Find Then, extend z = to find V(y 2 –z 2 – 1, 2z 4 -3z 2 +1) and so on. We will receive V(g 1, g 2,g 3 ) = By theorem, V(g 1, g 2,..., g t ) = V( f 1, f 2,..., f s ), which is the set of all solutions Buchberger’s Algorithm Input: F = (f 1, …, f s ) Output: a Gröbner basis for I = with G := F REPEAT G’ := G FOR in G’ DO S := UNTIL G = G’ Solving Method Prof. B. Buchberger