Resultants for Multivariate Polynomials7/14/2008, 14.00-15.30Manfred MinimairE-mail: manfred@minimair.orgDepartment of Mathematics and Computer ScienceSeton Hall University, South Orange, New Jersey, USAMultivariate resultantsProblem statementBasic casesMacaulay's constructionCanny's generalized characteristic polynomialsProblems arising from Macaulay's constructionDixon resultant constructionTypical uses of multivariate resultantsReferences: Cox/Little/O'Shea, "Ideals, Varieties, and Algorithms", "Using Algebraic Geometry"Problem statementInput: polynomials f0, f1, ..., fn in variables x1, ..., xn, of degrees d0, d1, ..., dn.Output: resultant r of f0, f1, ..., fn (w.r.t. degrees d0, d1, ..., dn) such that Vanishing Theorem holds.Vanishing Theoremf0, f1, ..., fn have a common root or their leading forms have non-trivial common root (not equal to (0,...,0))
iff thre resultant r vanishesThe resultant is an irreducible polynomial in the coefficients of f0, f1, ..., fn.Remark: leading forms= part of a polynomial containing monomials of highest degreeE.g.QyQ+SSJmRzYiLUklc29ydEclKnByb3RlY3RlZEc2JC1JKXJhbmRwb2x5R0YlNiU3JEkjeDFHRiVJI3gyR0YlL0knZGVncmVlR0YoIiIjSSZkZW5zZUdGJUYtIiIiprint();QyQtSSljb2VmdGF5bEc2IjYlSSJmR0YlLzckSSN4MUdGJUkjeDJHRiU3JCIiIUYtNyRGLSIiIyIiIg==leading form of f: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QyQsKComLUknbWNvZWZmRzYiNiVJImZHRic3JEkjeDFHRidJI3gyR0YnKiQpRisiIiMiIiJGMEYuRjBGMCooLUYmNiVGKTckRitGKyomRitGMEYsRjBGMEYrRjBGLEYwRjAqJi1GJjYlRilGKiokKUYsRi9GMEYwRjpGMEYwRjA=Basic casesDegrees of polynomials are all 1.E.g.QyQ+SSNmMEc2IiwoKiZJI2ExR0YlIiIiSSN4MUdGJUYpRikqJkkjYTJHRiVGKUkjeDJHRiVGKUYpSSNhM0dGJUYpRik=QyQ+SSNmMUc2IiwoKiZJI2IxR0YlIiIiSSN4MUdGJUYpRikqJkkjYjJHRiVGKUkjeDJHRiVGKUYpSSNiM0dGJUYpRik=QyQ+SSNmMkc2IiwoKiZJI2MxR0YlIiIiSSN4MUdGJUYpRikqJkkjYzJHRiVGKUkjeDJHRiVGKUYpSSNjM0dGJUYpRik=What is a condition for f0, f1, f2 to have a common root?Idea: view f0=0, f1=0, f2=0 as linear system:QyQvLCgqJkkjYTFHNiIiIiJJI3gxR0YnRihGKComSSNhMkdGJ0YoSSN4MkdGJ0YoRigqJkkjYTNHRidGKEkiekdGJ0YoRigiIiFGKA==QyQvLCgqJkkjYjFHNiIiIiJJI3gxR0YnRihGKComSSNiMkdGJ0YoSSN4MkdGJ0YoRigqJkkjYjNHRidGKEkiekdGJ0YoRigiIiFGKA==QyQvLCgqJkkjYzFHNiIiIiJJI3gxR0YnRihGKComSSNjMkdGJ0YoSSN4MkdGJ0YoRigqJkkjYzNHRidGKEkiekdGJ0YoRigiIiFGKA==If system has a non-trivial root (x1, x2, 1) then determinant of coefficient matrix vanishes.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QyQ+SSJyRzYiLV9JLkxpbmVhckFsZ2VicmFHRiVJLERldGVybWluYW50R0YlNiNJIk1HRiUiIiI=It can be shown that this determinant is the resultant of f0, f1, f2.Resultant also vanishes if leading forms a1*x1+a2*x2, b1*x1+b2*x2, c1*x1+c2*x2 have a common non-trivial root.
E.g. a1=b1=c1, a2=b2=c2QyQtSSVzdWJzRyUqcHJvdGVjdGVkRzYkNyYvSSNiMUc2IkkjYTFHRiovSSNjMUdGKkYrL0kjYjJHRipJI2EyR0YqL0kjYzJHRipGMEkickdGKiIiIg==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnMacaulay's algorithm for constructing resultantsConstruct Macaulay matrix from f0, f1, ..., fnDetermine a certain submatrix of Macaulay matrixDivide determinant of Macaulay matrix by determinant of submatrixLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnExample for degree 2 polynomialsQyRBUSdtci5tcGw2IiEiIg==QyQ+SSNmMEc2IiwuKiZJJGEyMEdGJSIiIilJI3gwR0YlIiIjRilGKSooSSRhMTFHRiVGKUYrRilJI3gxR0YlRilGKSooSSRhMTBHRiVGKUYrRilJI3gyR0YlRilGKSomSSRhMDJHRiVGKSlGL0YsRilGKSooSSRhMDFHRiVGKUYvRilGMkYpRikqJkkkYTAwR0YlRikpRjJGLEYpRikhIiI=QyQ+SSNmMUc2IiwuKiZJJGIyMEdGJSIiIilJI3gwR0YlIiIjRilGKSooSSRiMTFHRiVGKUYrRilJI3gxR0YlRilGKSooSSRiMTBHRiVGKUYrRilJI3gyR0YlRilGKSomSSRiMDJHRiVGKSlGL0YsRilGKSooSSRiMDFHRiVGKUYvRilGMkYpRikqJkkkYjAwR0YlRikpRjJGLEYpRikhIiI=QyQ+SSNmMkc2IiwoKiZJJGMxMEdGJSIiIkkjeDBHRiVGKUYpKiZJJGMwMUdGJUYpSSN4MUdGJUYpRikqJkkkYzAwR0YlRilJI3gyR0YlRilGKSEiIg==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QyQtX0kuTGluZWFyQWxnZWJyYUc2JEkoX3N5c2xpYkc2IiUqcHJvdGVjdGVkR0krRGltZW5zaW9uc0dGKDYjSSJNR0YoIiIiThe Macaulay matrix is the coefficient matrix of multples of f0, f1, f2. The multipliers are power products of x0, x1, x2 with the following exponents.QyRJKmV4cG9uZW50c0c2IiIiIg==Here, respectively, multiplied with f0, f0, f0, f1, f1, f1, f2, f2, f2, f2, as can be seen from M.The submatrix consists of so-called unreduced rows and columns.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QyZJKnVucmVkdWNlZEc2IiIiIi1JJHNlcUclKnByb3RlY3RlZEc2JCZJKmV4cG9uZW50c0dGJDYjSSJpR0YkLUkjaW5HRig2JEYtRiNGJQ==QyQ2JCZJIk1HNiI2JCIiJEYoJkYlNiRGKCIiJyIiIg==QyQ2JCZJIk1HNiI2JCIiJyIiJCZGJTYkRihGKCIiIg==Thus the resultant isLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnQyQ+SSRkZXRHNiJfSS5MaW5lYXJBbGdlYnJhRzYkSShfc3lzbGliR0YlJSpwcm90ZWN0ZWRHSSxEZXRlcm1pbmFudEdGJSEiIg==QyQ+SSJyRzYiLUknbm9ybWFsRyUqcHJvdGVjdGVkRzYjKiYtSSRkZXRHRiU2I0kiTUdGJSIiIi1GLDYjSSJTR0YlISIiRjM=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn1 Construct Macaulay matrixdd = d0+...+dn -nQyQ+SSNkZEc2Ii1JIitHJSpwcm90ZWN0ZWRHNiQsJi1GJzYkIiIjRi0iIiJGLkYuISIjRi4=S0 = {monomials in x0,...,xn of degree dd such that divisible by x0^d0}
S1 = {monomials in x0,...,xn of degree dd such that not divisible by x0^d0 but by x1^d1}...Sn = {monomials in x0,...,xn of degree dd such that not divisible by x0^d0,...,x_n-1^d_n-1 but by xn^dn}LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnMonomials for exampleQyY+SSJURzYiNywqJClJI3gwR0YlIiIkIiIiKiQpSSN4MUdGJUYqRisqJClJI3gyR0YlRipGKyomKUYpIiIjRitGLkYrKiZGM0YrRjFGKyomKUYuRjRGK0YpRisqJkY3RitGMUYrKiYpRjFGNEYrRilGKyomRjpGK0YuRisqKEYpRitGLkYrRjFGK0YrLUklbm9wc0clKnByb3RlY3RlZEc2I0YkRis=QyQ+SSNTMEc2IjclKiQpSSN4MEdGJSIiJCIiIiomKUYpIiIjRitJI3gxR0YlRisqJkYtRitJI3gyR0YlRitGKw==QyQ+SSNTMUc2IjclKiQpSSN4MUdGJSIiJCIiIiomKUYpIiIjRitJI3gwR0YlRisqJkYtRitJI3gyR0YlRitGKw==QyQ+SSNTMkc2IjcmKiQpSSN4MkdGJSIiJCIiIiomKUYpIiIjRitJI3gwR0YlRisqJkYtRitJI3gxR0YlRisqKEYvRitGMUYrRilGK0YrThen take coefficient matrix of S0/x0^d0*f0, S1/x1^d1*f1, S2/x2^d2*f2.QyQtSSRzZXFHJSpwcm90ZWN0ZWRHNiQtSSdleHBhbmRHRiU2IyooJkkjUzBHNiI2I0kiaUdGLSIiIilJI3gwR0YtIiIjISIiSSNmMEdGLUYwL0YvO0YwLUklbm9wc0dGJTYjRixGMA==QyQtSSRzZXFHJSpwcm90ZWN0ZWRHNiQtSSdleHBhbmRHRiU2IyooJkkjUzFHNiI2I0kiaUdGLSIiIilJI3gxR0YtIiIjISIiSSNmMUdGLUYwL0YvO0YwLUklbm9wc0dGJTYjRixGMA==QyQtSSRzZXFHJSpwcm90ZWN0ZWRHNiQtSSdleHBhbmRHRiU2IyooJkkjUzJHNiI2I0kiaUdGLSIiIkkjeDJHRi0hIiJJI2YyR0YtRjAvRi87RjAtSSVub3BzR0YlNiNGLEYwLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn2 SubmatrixDefinition of reducedA monomial of degree dd is reduced iff divisible by exactly one xi^di.unreduced = not reduced (divisible by more than one xi^di or none)Find submatrix of rows/columns corresponding to unreduced monomial multipliers.QyQ+SSp1bnJlZHVjZWRHNiI3JComKUkjeDBHRiUiIiMiIiJJI3gyR0YlRisqJilJI3gxR0YlRipGK0YsRitGKw==unreduced[1] in S0 and unreduced[2] in S1Take rows ofQyQ+SSNoMUc2Ii1JJXNvcnRHJSpwcm90ZWN0ZWRHNiQtSSdleHBhbmRHRig2IyomLUkiKkdGKDYkKiYpSSN4MEdGJSIiIyIiIkkjeDJHRiVGNSokRjIhIiJGNUkjZjBHRiVGNTclRjNJI3gxR0YlRjZGNQ==QyQ+SSNoMkc2Ii1JJXNvcnRHJSpwcm90ZWN0ZWRHNiQtSSdleHBhbmRHRig2IyomLUkiKkdGKDYkKiYpSSN4MUdGJSIiIyIiIkkjeDJHRiVGNSokRjIhIiJGNUkjZjFHRiVGNTclSSN4MEdGJUYzRjZGNQ==Columns are coefficients of unreduced monomials:QyQ2JC1JKWNvZWZ0YXlsRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiVJI2gxR0YpLzclSSN4MEdGKUkjeDFHRilJI3gyR0YpNyUiIiFGMkYyNyUiIiNGMiIiIi1GJTYlRitGLDclRjJGNEY1RjU=QyQ2JC1JKWNvZWZ0YXlsRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiVJI2gyR0YpLzclSSN4MEdGKUkjeDFHRilJI3gyR0YpNyUiIiFGMkYyNyUiIiNGMiIiIi1GJTYlRitGLDclRjJGNEY1RjU=3 Divide determinant of Macaulay matrix by determinant of submatrixLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnProblem: What to do when determinant of submatrix vanishes?LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnCanny's generalized characteristic polynomialLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnExampleLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnQyQ+SSNmMEc2IiwuKiZJJGEyMEdGJSIiIilJI3gwR0YlIiIjRilGKSooSSRhMTFHRiVGKUYrRilJI3gxR0YlRilGKSooSSRhMTBHRiVGKUYrRilJI3gyR0YlRilGKSomSSRhMDJHRiVGKSlGL0YsRilGKSooSSRhMDFHRiVGKUYvRilGMkYpRikqJkkkYTAwR0YlRikpRjJGLEYpRikhIiI=QyQ+SSNmMUc2IiwuKihJImRHRiUiIiJJJGEyMEdGJUYpKUkjeDBHRiUiIiNGKUYpKihJJGIxMUdGJUYpRixGKUkjeDFHRiVGKUYpKihJJGIxMEdGJUYpRixGKUkjeDJHRiVGKUYpKihGKEYpSSRhMDJHRiVGKSlGMEYtRilGKSooSSRiMDFHRiVGKUYwRilGM0YpRikqJkkkYjAwR0YlRikpRjNGLUYpRikhIiI=QyQ+SSNmMkc2IiwoKiZJJGMxMEdGJSIiIkkjeDBHRiVGKUYpKiZJJGMwMUdGJUYpSSN4MUdGJUYpRikqJkkkYzAwR0YlRilJI3gyR0YlRilGKSEiIg==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QyQtSSRkZXRHNiI2I0kiU0dGJSIiIg==Still, resultant exists:QyQ+SSJyRzYiLV9JI01SR0YlSStNUmVzdWx0YW50R0YlNiQtSSVzdWJzRyUqcHJvdGVjdGVkRzYkL0kjeDJHRiUiIiI3JUkjZjBHRiVJI2YxR0YlSSNmMkdGJTckSSN4MEdGJUkjeDFHRiVGMQ==Solution: infinitesimal perturbation1) divide characteristic polynomials of Macaulay matrix and submatrix -> "Canny's generalized characteristic polynomial"2) resultant is constant term of generalized characteristic polynomialQyQ+SSljaGFycG9seUc2Il9JLkxpbmVhckFsZ2VicmFHNiRJKF9zeXNsaWJHRiUlKnByb3RlY3RlZEdJOUNoYXJhY3RlcmlzdGljUG9seW5vbWlhbEdGJSEiIg==QyQ+SSNjbUc2Ii1JKWNoYXJwb2x5R0YlNiRJIk1HRiVJJ2xhbWJkYUdGJSEiIg==QyQ+SSNjc0c2Ii1JKWNoYXJwb2x5R0YlNiRJIlNHRiVJJ2xhbWJkYUdGJSIiIg==QyQ+SSVjbWNzRzYiLUknbm9ybWFsRyUqcHJvdGVjdGVkRzYjKiZJI2NtR0YlIiIiSSNjc0dGJSEiIkYuQyQ+SSJSRzYiLUklc3Vic0clKnByb3RlY3RlZEc2JC9JJ2xhbWJkYUdGJSIiIUklY21jc0dGJSEiIg==QyQtSSZldmFsYkclKnByb3RlY3RlZEc2Iy9JIlJHNiJJInJHRikiIiI=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnWhy "characteristic polynomial"?QyQ+SSNoMEc2IiwuKiYsJkkkYTIwR0YlIiIiSSdsYW1iZGFHRiUhIiJGKilJI3gwR0YlIiIjRipGKiooSSRhMTFHRiVGKkYuRipJI3gxR0YlRipGKiooSSRhMTBHRiVGKkYuRipJI3gyR0YlRipGKiomSSRhMDJHRiVGKilGMkYvRipGKiooSSRhMDFHRiVGKkYyRipGNUYqRioqJkkkYTAwR0YlRiopRjVGL0YqRipGLA==QyQ+SSNoMUc2IiwuKihJImRHRiUiIiJJJGEyMEdGJUYpKUkjeDBHRiUiIiNGKUYpKihJJGIxMUdGJUYpRixGKUkjeDFHRiVGKUYpKihJJGIxMEdGJUYpRixGKUkjeDJHRiVGKUYpKiYsJiomRihGKUkkYTAyR0YlRilGKUknbGFtYmRhR0YlISIiRikpRjBGLUYpRikqKEkkYjAxR0YlRilGMEYpRjNGKUYpKiZJJGIwMEdGJUYpKUYzRi1GKUYpRjk=QyQ+SSNoMkc2IiwoKiZJJGMxMEdGJSIiIkkjeDBHRiVGKUYpKiZJJGMwMUdGJUYpSSN4MUdGJUYpRikqJiwmSSRjMDBHRiVGKUknbGFtYmRhR0YlISIiRilJI3gyR0YlRilGKUYyQyQ+SSNyaEc2Ii1JJXN1YnNHJSpwcm90ZWN0ZWRHNiQvSSdsYW1iZGFHRiUiIiEtX0kjTVJHRiVJK01SZXN1bHRhbnRHRiU2JC1GJzYkL0kjeDJHRiUiIiI3JUkjaDBHRiVJI2gxR0YlSSNoMkdGJTckSSN4MEdGJUkjeDFHRiUhIiI=QyQtSSZldmFsYkclKnByb3RlY3RlZEc2Iy8tSSlzaW1wbGlmeUc2IjYjLCZJInJHRioiIiJJI3JoR0YqISIiIiIhRi4=Problems arising from Macaulay's constructionThere are parametric polynomials for which the Macaulay resultant construction vanishes for any parameters even though the polynomials do not have a common root in general.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Why?Leading forms a11*x1*x2, b11*x1*x2, c11*x1*x2 have non-trivial common roots, e.g. (1,0) or (0,1).Dixon resultant construction often allows to avoid this problem (ignores certain trivial roots of leading forms).QyRBUSdkci5tcGw2IiEiIg==QyQtX0kjRFJHNiJJK0RSZXN1bHRhbnRHRiY2JDclSSNmMEdGJkkjZjFHRiZJI2YyR0YmNyRJI3gxR0YmSSN4MkdGJiIiIg==Dixon resultant construction(generalization of Bezout resultant construction for univariate polynomials)Dixon/Bezout/Caley/Kapur/Yang/Saxena algorithm1 Construct Dixon polynomial from polynomials f0, f1, ..., fn in variables x1, ..., xn2 Construct Dixon matrix, the coefficient matrix of the Dixon polynomial3 Compute determinant of a maximal minor of the Dixon matrix--> The determinant is a multiple of the resultant of f0, f1, ..., fnExample for algorithmQyQ+SSNmMEc2IiwqKihJJGExMUdGJSIiIkkjeDFHRiVGKUkjeDJHRiVGKUYpKiZJJGExMEdGJUYpRipGKUYpKiZJJGEwMUdGJUYpRitGKUYpSSRhMDBHRiVGKUYpQyQ+SSNmMUc2IiwqKihJJGIxMUdGJSIiIkkjeDFHRiVGKUkjeDJHRiVGKUYpKiZJJGIxMEdGJUYpRipGKUYpKiZJJGIwMUdGJUYpRitGKUYpSSRiMDBHRiVGKUYpQyQ+SSNmMkc2IiwqKihJJGMxMUdGJSIiIkkjeDFHRiVGKUkjeDJHRiVGKUYpKiZJJGMxMEdGJUYpRipGKUYpKiZJJGMwMUdGJUYpRitGKUYpSSRjMDBHRiVGKUYpDixon polynomialLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzY+LUkjbWlHRiQ2JVEibUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNjBRIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUlZm9ybUdRIUYnLyUnbHNwYWNlR1EkMGVtRicvJSdyc3BhY2VHRk8vJShtaW5zaXplR1EiMUYnLyUobWF4c2l6ZUdRKWluZmluaXR5RictRjY2MFEqJmNvbG9uZXE7RidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSZpbmZpeEYnL0ZOUS90aGlja21hdGhzcGFjZUYnL0ZRRmhuRlJGVUY1LUYsNiVRJ01hdHJpeEYnRi9GMi1JKG1mZW5jZWRHRiQ2JC1GIzYjLUZebzYmLUYjNjotRiw2JkZMRi8vJStleGVjdXRhYmxlR0Y9RjItSSdtc3BhY2VHRiQ2Ji8lJ2hlaWdodEdRJjAuMGV4RicvJSZ3aWR0aEdRJjAuMGVtRicvJSZkZXB0aEdGX3AvJSpsaW5lYnJlYWtHUShuZXdsaW5lRictRl5vNictRiM2KS1GLDYmUSNmMEYnRi9GaG9GMi1GNjYxUSIsRidGaG9GOUY7L0Y/RjFGQEZCRkRGRkZIRmVuRk0vRlFRM3Zlcnl0aGlja21hdGhzcGFjZUYnRlJGVS1GNjYxRjhGaG9GOUY7Rj5GQEZCRkRGRkZIRkpGTUZQRlJGVS1GLDYmUSNmMUYnRi9GaG9GMkZfcUZlcS1GLDYmUSNmMkYnRi9GaG9GMkZob0Y5LyUlb3BlbkdRIltGJy8lJmNsb3NlR1EiXUYnRl9xRmpvLUYsNiZRJXN1YnNGJ0YvRmhvRjItRl5vNiUtRiM2JS1GXm82Jy1GIzYlLUYsNiZRI3gxRidGL0Zob0YyLUY2NjFRIj1GJ0Zob0Y5RjtGPkZARkJGREZGRkhGZW5GZ25GaW5GUkZVLUYsNiZRI3oxRidGL0Zob0YyRmhvRjlGXXJGYHJGX3EtRl5vNictRiM2J0ZccUZfcUZncUZfcUZqcUZob0Y5Rl1yRmByRmhvRjlGX3FGZXEtRjY2MVEiI0YnRmhvRjlGO0Y+RkBGQkZERkZGSEZKRk1GUEZSRlUtRiw2JlEocmVwbGFjZUYnRi9GaG9GMkZlcUZec0ZlcS1GLDYmUSV3aXRoRidGL0Zob0YyRmVxLUYsNiZRJG5ld0YnRi9GaG9GMkZlcS1GLDYmUSl2YXJpYWJsZUYnRi9GaG9GMkZlcUZkc0Zqb0Zjci1GXm82JS1GIzYlLUZebzYnLUYjNilGXnNGYXNGZHNGX3EtRiw2JlEjeDJGJ0YvRmhvRjJGYXMtRiw2JlEjejJGJ0YvRmhvRjJGaG9GOUZdckZgckZfcUZnc0Zob0Y5RjlGXXJGYHJGOS1GNjYxUSI7RidGaG9GOUY7RmJxRkBGQkZERkZGSEZlbkZNRmluRlJGVUZlcUZbdEZedEZlcUZec0ZlcS1GNjYxUSRhbmRGJ0Zob0Y5RjtGPkZARkJGREZGRkhGSkZNRlBGUkZVRmVxRmJ1RmVxRmF0RmVxRmR0RmVxLUYsNiZRKnZhcmlhYmxlc0YnRi9GaG9GMkZlcUZkc0ZlcUZbdkZlcUZldQ==QyQ+SSNkcEc2Ii1JJ25vcm1hbEclKnByb3RlY3RlZEc2IyooLV9JLkxpbmVhckFsZ2VicmFHRiVJLERldGVybWluYW50R0YlNiNJIm1HRiUiIiIsJkkjeDFHRiVGMUkjejFHRiUhIiJGNSwmSSN4MkdGJUYxSSN6MkdGJUY1RjVGMQ==Dixon matrixcoefficient matrix with respect to monomials in z1, z2 (rows) and x1, x2 (columns)LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYrLUkjbWlHRiQ2JVEjRE1GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYwUSJ+RicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJWZvcm1HUSFGJy8lJ2xzcGFjZUdRJDBlbUYnLyUncnNwYWNlR0ZPLyUobWluc2l6ZUdRIjFGJy8lKG1heHNpemVHUSlpbmZpbml0eUYnLUY2NjBRKiZjb2xvbmVxO0YnRjlGO0Y+RkBGQkZERkZGSC9GS1EmaW5maXhGJy9GTlEvdGhpY2ttYXRoc3BhY2VGJy9GUUZobkZSRlVGNS1GLDYlUSNEUkYnRi9GMi1GNjYwUSM6LUYnRjlGO0Y+RkBGQkZERkZGSEZKRk1GUEZSRlUtRiw2JVEoRE1hdHJpeEYnRi9GMi1JKG1mZW5jZWRHRiQ2JC1GIzYtLUZkbzYmLUYjNictRiw2JVEjZjBGJ0YvRjItRjY2MFEiLEYnRjlGOy9GP0YxRkBGQkZERkZGSEZlbkZNL0ZRUTN2ZXJ5dGhpY2ttYXRoc3BhY2VGJ0ZSRlUtRiw2JVEjZjFGJ0YvRjJGX3AtRiw2JVEjZjJGJ0YvRjJGOS8lJW9wZW5HUSJbRicvJSZjbG9zZUdRIl1GJ0ZfcC1GZG82Ji1GIzYlLUYsNiVRI3gxRidGL0YyRl9wLUYsNiVRI3gyRidGL0YyRjlGW3FGXnFGX3AtRjY2MFEiJ0YnRjlGO0Y+RkBGQkZERkZGSC9GS1EocG9zdGZpeEYnL0ZOUTJ2ZXJ5dGhpbm1hdGhzcGFjZUYnRlBGUkZVLUYsNiVRKHJvd01vbnNGJ0YvRjJGW3JGX3BGW3ItRiw2JVEoY29sTW9uc0YnRi9GMkZbckY5LUY2NjBRIjtGJ0Y5RjtGYnBGQEZCRkRGRkZIRmVuRk1GaW5GUkZVQyRJKHJvd01vbnNHNiIiIiI=(x1_ corresponds to z1 above. x1_ is the new variable symbol automatically chosen by DR:-DMatrix.)QyRJKGNvbE1vbnNHNiIiIiI=Determinant of maximal minorHere DM is square. Therefore simply compute the determinant.QyQ+SSJyRzYiLV9JLkxpbmVhckFsZ2VicmFHNiRJKF9zeXNsaWJHRiUlKnByb3RlY3RlZEdJLERldGVybWluYW50R0YlNiNJI0RNR0YlISIiTypical uses of multivariate resultants1 Find parameters such that parametric multivariate polynomials have a common root(compare univariate examples)2 Triangularize systems of equations (see also univariate example)ExamplePkkjZjBHNiItSSVzb3J0RyUqcHJvdGVjdGVkRzYkLUkpcmFuZHBvbHlHNiRGJ0koX3N5c2xpYkdGJDYlNyVJI3gxR0YkSSN4MkdGJEkjeDNHRiQvSSdkZWdyZWVHRiciIiNJJmRlbnNlR0YkRi4=PkkjZjFHNiItSSVzb3J0RyUqcHJvdGVjdGVkRzYkLUkpcmFuZHBvbHlHNiRGJ0koX3N5c2xpYkdGJDYlNyVJI3gxR0YkSSN4MkdGJEkjeDNHRiQvSSdkZWdyZWVHRiciIiNJJmRlbnNlR0YkRi4=PkkjZjJHNiItSSVzb3J0RyUqcHJvdGVjdGVkRzYkLUkpcmFuZHBvbHlHNiRGJ0koX3N5c2xpYkdGJDYlNyVJI3gxR0YkSSN4MkdGJEkjeDNHRiQvSSdkZWdyZWVHRiciIiJJJmRlbnNlR0YkRi4=Eliminate x1 and x2QyQ+SSRyeDNHNiItX0kjTVJHRiVJK01SZXN1bHRhbnRHRiU2JDclSSNmMEdGJUkjZjFHRiVJI2YyR0YlNyRJI3gxR0YlSSN4MkdGJSIiIg==Now, possible to solve for x3.Eliminate x1QyQ+SSZyeDJ4M0c2Ii1JKnJlc3VsdGFudEdGJTYlSSNmMEdGJUkjZjFHRiVJI3gxR0YlIiIiPlug in solutions x3 and solve for x2. Then solve for x1, e.g, using f0.Finally check if computed roots of f0 are also roots of f1 and f2.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn