Solving A System of Polynomial Equations

1. Solving A System of Polynomial Equations Narapong Srivisal, Swarthmore College Class of 2007 Solving Method A monomial in x1,x2,…,xnis a product of the form ,where each is a non-negative integer. The total degreeof 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[x1, …, xn] for the set of all polynomials in x1, x2, …, xnwith coefficients in k. DefinitionLexicographic Order: Fix variable order x1 > x2 > … > xn.Let . Then, if the leftmost entry of is positive. The leading termof with respect to a monomial order >, LT>(f), is ,where is the largest monomial appearing in f in the ordering >, and xα is the leading monomial. DefinitionAffine Variety V(f1,…, fs) TheoremIf , then V(f1,…, fs) = V(g1,…, gt). DefinitionLet . Themth elimination ideal Imis the ideal of defined by TheoremIm is an ideal for of k[xm+1, xm+2, …, xn] • Let f1 = f2 = f3 = … = fs = 0 be the system of polynomial equations in • that we would like to solve • eg. in , 0 = x2+ y2 + z2 – 4 = f1 • 0 = x2 + 2y2 – 5 = f2 • 0 = xz – 1 = f3 • The set of all solutions of f1 =f2 = f3 = … = fs = 0 is V(f1, f2, f3, …, fs) • Change the basis for I to a Gröbner basis, G = {g1, g2,g3, ..., gt}, 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, Gm = is a Gröbner basis for • the mth elimination ideal Im. 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(Im), and extend to V(Im-1), starting • with m = n – 1 until receiving the complete solutions, V(g1, g2,..., gt). • eg. Find • Then, extend z = to find V(y2 –z2 – 1, 2z4 -3z2 +1) and so on. • We will receive V(g1, g2,g3) = • By theorem, V(g1, g2,..., gt) = V(f1, f2, ..., fs), which is the set of all solutions The Elimination Theorem Let G be a Gröbner basis for I with respect to lexicographic order, where x1 > x2 > … > xn. Then, for every 0 ≤ m ≤ n, the set is a Gröbner basis for the mth elimination ideal Im Proof Let G = {g1,..., gt}. Fix 0 ≤ m ≤ n. Suppose Gm = {g1, ..., gr}, where r < t i). WTS Gm is a basis for Im + Since and Im is an ideal, then + Pick any . By Division Algorithm and theorem, f = h1g1 + h2g2 + … + hrgr + h(r + 1)g(r+1) + … + htgt However, for each i > r, gi has at least one term involving x1, x2, …, or xm. Thus, , . Therefore, , by Division Algorithm. ii). WTS Gm is a Gröbner basis for Im Let 1≤ i< j≤ r. Since , so are the leading terms and the l.c.m. monomail of the leading terms, . Then, Therefore, , and Gm is a Gröbner basis for Imby theorem. By i and ii, the theorem has been proved. The Extension Theorem Let , where k is an algebraically closed field Let I1 be the first elimination ideal of I. Write where Ni ≥ 0 and is non-zero. Suppose (a2, a3, … , an) . If (a2, a3, … , an) , then Buchberger’s Algorithm Input: F = (f1, … , fs) Output: a Gröbner basis for I = with G := F REPEAT G’ := G FOR in G’ DO S := UNTIL G = G’ Division Algorithm Fix a monomial order > in k[x1, …, xn]. Let F = (f1, … , fs) be an ordered s-tuple of polynomials in k[x1, …, xn]. Then every can be written as f = h1f1+ … + hsfs + where hi, . . For all i, hifi = 0 or , and is equal to zero or a linear combination of monomials, non of which is divisible by LT>(fi),…, LT>(fs). Call a remainder of f on division by F Prof. B. Buchberger Gröbner Basis Let be an ideal. A Gröbner basisfor I is a finite collection of polynomials s.t. , LT(f) is divisible by LT(gi) 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’. TheoremA Gröbner basis for I is a basis for I. DefinitionS-Polynomialof , denoted S(f,g), is the polynomial where is the least common multiple monomial of LT(f) and LT(g). References Cox, David A. and John Little and Donal O’Shea. (2004) Using Algebraic Geometry (2nd 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 Theorem (Buchberger’s Criterion) is a Gröbner basis for iff, .