LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYvLUkjbW9HRiQ2MFEifkYnLyUlc2l6ZUdRIzE2RicvJSVib2xkR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSVib2xkRicvJStmb250d2VpZ2h0R0Y3LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y8LyUpc3RyZXRjaHlHRjwvJSpzeW1tZXRyaWNHRjwvJShsYXJnZW9wR0Y8LyUubW92YWJsZWxpbWl0c0dGPC8lJ2FjY2VudEdGPC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRktGKy1JI21pR0YkNihRLEFsdGVybmF0aW5nRidGL0YyLyUnaXRhbGljR0Y8RjVGOEYrLUZPNihRLFBvbHlub21pYWxzRidGL0YyRlJGNUY4RistRiw2MFEjYnlGJ0YvRjJGNUY4RjpGPUY/RkFGQ0ZFRkdGSUZMRistRk82KFEmTWFwbGVGJ0YvRjJGUkY1RjhGL0YyRjVGOA==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYvLUkjbW9HRiQ2MFEifkYnLyUlc2l6ZUdRIzE2RicvJSVib2xkR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSVib2xkRicvJStmb250d2VpZ2h0R0Y3LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y8LyUpc3RyZXRjaHlHRjwvJSpzeW1tZXRyaWNHRjwvJShsYXJnZW9wR0Y8LyUubW92YWJsZWxpbWl0c0dGPC8lJ2FjY2VudEdGPC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRktGK0YrLUkjbWlHRiQ2KFEnS2FodGFuRidGL0YyLyUnaXRhbGljR0Y8RjVGOEYrLUZPNihRIkhGJ0YvRjJGUkY1RjgtRiw2MFEiLkYnRi9GMkY1RjhGOkY9Rj9GQUZDRkVGR0ZJRkxGKy1GTzYoUSpBbHp1YmFpZHlGJ0YvRjJGUkY1RjhGL0YyRjVGOA==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbW9HRiQ2MFEifkYnLyUlc2l6ZUdRIzE2RicvJSVib2xkR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSVib2xkRicvJStmb250d2VpZ2h0R0Y3LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y8LyUpc3RyZXRjaHlHRjwvJSpzeW1tZXRyaWNHRjwvJShsYXJnZW9wR0Y8LyUubW92YWJsZWxpbWl0c0dGPC8lJ2FjY2VudEdGPC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRktGK0YvLyUrZXhlY3V0YWJsZUdGNC8lMGZvbnRfc3R5bGVfbmFtZUdRKTJEfklucHV0RicvRjZRJ25vcm1hbEYnLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzY9LUkjbW9HRiQ2L1EifkYnLyUlYm9sZEdRJXRydWVGJy8lLG1hdGh2YXJpYW50R1ElYm9sZEYnLyUrZm9udHdlaWdodEdGNC8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGOS8lKXN0cmV0Y2h5R0Y5LyUqc3ltbWV0cmljR0Y5LyUobGFyZ2VvcEdGOS8lLm1vdmFibGVsaW1pdHNHRjkvJSdhY2NlbnRHRjkvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZIRitGK0YrRitGKy1JI21pR0YkNihRJEtleUYnLyUlc2l6ZUdRIzE0RidGLy8lJ2l0YWxpY0dGOUYyRjUtRiw2MEYuRk9GL0YyRjVGN0Y6RjxGPkZARkJGREZGRkktRkw2KFEmV29yZHNGJ0ZPRi9GUkYyRjUtRiw2MFEiOkYnRk9GL0YyRjVGN0Y6RjxGPkZARkJGRC9GR1EsMC4yNzc3Nzc4ZW1GJy9GSkZnbkZULUZMNihRKlN5bW1ldHJpY0YnRk9GL0ZSRjJGNUZULUZMNihRK1BvbHlub21pYWxGJ0ZPRi9GUkYyRjUtRiw2MFEiLEYnRk9GL0YyRjVGNy9GO0YxRjxGPkZARkJGREZGL0ZKUSwwLjMzMzMzMzNlbUYnLUZMNihRLEFsdGVybmF0aW5nRidGT0YvRlJGMkY1RlRGXG9GVC1GLDYwUSRhbmRGJ0ZPRi9GMkY1RjdGOkY8Rj5GQEZCRkRGRkZJRlQtRkw2KFEpR3JvZWJuZXJGJ0ZPRi9GUkYyRjVGVC1GTDYoUSZCYXNpc0YnRk9GL0ZSRjJGNS8lK2V4ZWN1dGFibGVHRjEvJTBmb250X3N0eWxlX25hbWVHUSkyRH5JbnB1dEYnL0YzUSdub3JtYWxGJw==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2KFEtSW50cm9kdWN0aW9uRicvJSVib2xkR1EldHJ1ZUYnLyUnaXRhbGljR1EmZmFsc2VGJy8lKnVuZGVybGluZUdGMS8lLG1hdGh2YXJpYW50R1ElYm9sZEYnLyUrZm9udHdlaWdodEdGOS1JI21vR0YkNi9RIi5GJ0YvRjdGOi8lJmZlbmNlR0Y0LyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjQvJSpzeW1tZXRyaWNHRjQvJShsYXJnZW9wR0Y0LyUubW92YWJsZWxpbWl0c0dGNC8lJ2FjY2VudEdGNC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlAvJStleGVjdXRhYmxlR0YxLyUwZm9udF9zdHlsZV9uYW1lR1EpMkR+SW5wdXRGJy9GOFEnbm9ybWFsRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzZqby1JI21pR0YkNidRIkFGJy8lJWJvbGRHUSV0cnVlRicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSVib2xkRicvJStmb250d2VpZ2h0R0Y3LUkjbW9HRiQ2L1EifkYnRi9GNUY4LyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTi1GLDYnUS1tdWx0aXZhcmlhdGVGJ0YvRjJGNUY4RjotRiw2J1ErcG9seW5vbWlhbEYnRi9GMkY1RjhGOkY6LUYsNidRImZGJ0YvRjJGNUY4LUkobWZlbmNlZEdGJDYmLUYjNistSSVtc3ViR0YkNiUtRiw2J1EieEYnRi9GMkY1RjgtRiM2Ji1JI21uR0YkNiZRIjFGJ0YvRjVGOC8lK2V4ZWN1dGFibGVHRjEvJTBmb250X3N0eWxlX25hbWVHUSkyRH5JbnB1dEYnL0Y2USdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnLUY7Ni9RIixGJ0YvRjVGOEY+L0ZBRjFGQkZERkZGSEZKRkwvRlBRLDAuMzMzMzMzM2VtRictRjs2L1EjLi5GJ0YvRjVGOEY+RkBGQkZERkZGSEZKL0ZNUSwwLjIyMjIyMjJlbUYnRk8tRjs2L1EiLkYnRi9GNUY4Rj5GQEZCRkRGRkZIRkpGTEZPRl9wLUZqbjYlRlxvLUYjNiYtRiw2J1EibkYnRi9GMkY1RjhGZW9GZ29Gam9GXHBGZW9GZ29Gam9GL0Y1RjhGOi1GLDYnUSNpc0YnRi9GMkY1RjhGOkY6LUYsNidRLGFsdGVybmF0aW5nRidGL0YyRjVGOEY6LUY7Ni1GPUZqb0Y+RkBGQkZERkZGSEZKRkxGTy1GOzYvUSNpZkYnRi9GNUY4Rj5GQEZCRkRGRkZIRkpGTEZPRmpxRmpxRlctRmVuNiYtRiM2Ki1Gam42JUZcby1GIzYnLUYsNidRKiZlcHNpbG9uO0YnRi9GMkY1RjgtRmVuNiZGX29GL0Y1RjhGZW9GZ29Gam9GXHBGX3BGZXBGX3AtRmpuNiVGXG8tRiM2J0Znci1GZW42JkZfcUYvRjVGOEZlb0Znb0Zqb0ZccEZlb0Znb0Zqb0YvRjVGOC1GOzYvUSI9RidGL0Y1RjhGPkZARkJGREZGRkhGSi9GTVEsMC4yNzc3Nzc4ZW1GJy9GUEZmcy1GLDYnUSRzZ25GJ0YvL0YzRjEvRjZRLGJvbGQtaXRhbGljRidGOC1GZW42Jy1GIzYpLUYsNidRKCYjMTAxMztGJ0YvRlt0Rlx0RjhGL0ZbdEZlb0Znb0ZcdEY4Ri9GW3RGXHRGOC1GLDYnRllGL0ZbdEZcdEY4RlpGOkZfcC1GLDYnUSZ3aGVyZUYnRi9GMkY1RjhGOkY6LUYsNidGanNGL0YyRjVGOC1GZW42Ji1GIzYoLUYsNidGZHRGL0YyRjVGOEYvRmVvRmdvRjVGOEYvRjVGOEZic0Zhb0Y6RlxyRmpxRmpxRmB1RjpGZHFGOi1GLDYnUSVldmVuRidGL0YyRjVGOEZqcS1GOzYvUSRhbmRGJ0YvRjVGOEY+RkBGQkZERkZGSEZKRkxGT0ZqcUZicy1GOzYvUSomdW1pbnVzMDtGJ0YvRjVGOEY+RkBGQkZERkZGSEZKRmhwL0ZQRmlwRmFvRmpxRlxyRmpxLUYsNiVGZHRGMkZqb0ZqcUZkcUY6LUYsNidRJG9kZEYnRi9GMkY1RjgtRjs2LUZccUZqb0Y+RkBGQkZERkZGSEZKRkxGT0ZqcS1GOzYvUSRmb3JGJ0YvRjVGOEY+RkBGQkZERkZGSEZKRkxGT0Y6LUYsNidRJGFueUYnRi9GMkY1RjhGOi1GLDYnUSxwZXJtdXRhdGlvbkYnRi9GMkY1RjhGOkZgdUY6LUYsNidRI29mRidGL0YyRjVGOEY6LUYsNidRJ2RlZ3JlZUYnRi9GMkY1RjhGOkY6RmFxRmpwRmVvRmdvRmpvThe polynomial is called symmetric, if 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 The basic example of alternating polynomials is the Vandermonde polynomial which is defined as follows : \316\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 Any alternating polynomial is a product of the Vandermonde polynomial and a symmetric polynomial.Fundamental Theorem of Symmetric Polynomials Any symmetric polynomial is a polynomial in elementary symmetric polynomials.LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2KFErUmVmZXJlbmNlc0YnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSp1bmRlcmxpbmVHRi4vJSxtYXRodmFyaWFudEdRJWJvbGRGJy8lK2ZvbnR3ZWlnaHRHRjY= [1] K.H.Alzubaidy, Symmetric Polynomials by Maple, Maple Application Center, Abstract Algebra, June 8, 2105.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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEld2l0aEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYlLUYsNiVRKWNvbWJpbmF0RidGL0YyLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvRjNRJ25vcm1hbEYnRkAtSSNtb0dGJDYtUSI6RidGQC8lJmZlbmNlR0Y/LyUqc2VwYXJhdG9yR0Y/LyUpc3RyZXRjaHlHRj8vJSpzeW1tZXRyaWNHRj8vJShsYXJnZW9wR0Y/LyUubW92YWJsZWxpbWl0c0dGPy8lJ2FjY2VudEdGPy8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlZGPUZAJSFHTo generate power sum polynomials arguments : list of variables L and power r.psp:=proc(L,r);add((L[i])^r,i=1..nops(L));end proc;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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=To generate elementary symmetric polynomials Elementary symmetric polynomials in 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 arguments : list of variables L and index r.esp:=proc(L,r)local f,i;f:=product((x_-L[i]),i=1..nops(L));simplify(coeff(f,x_,nops(L)-r))*(-1)^r;end proc;ExamplesLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEkZXNwRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYkLUYjNictRjY2Ji1GIzYpLUYsNiVRInhGJ0YvRjItSSNtb0dGJDYtUSIsRicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRkkvJSpzeW1tZXRyaWNHRkkvJShsYXJnZW9wR0ZJLyUubW92YWJsZWxpbWl0c0dGSS8lJ2FjY2VudEdGSS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYsNiVRInlGJ0YvRjJGQS1GLDYlUSJ6RidGL0YyLyUrZXhlY3V0YWJsZUdGSUZFRkUvJSVvcGVuR1EiW0YnLyUmY2xvc2VHUSJdRidGQS1JI21uR0YkNiRRIjBGJ0ZFRlxvRkVGRS1GQjYtUSI7RidGRUZHRkpGTEZORlBGUkZURlYvRlpRLDAuMjc3Nzc3OGVtRidGXG9GRQ==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEkZXNwRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYkLUYjNictRjY2Ji1GIzYpLUYsNiVRInhGJ0YvRjItSSNtb0dGJDYtUSIsRicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRkkvJSpzeW1tZXRyaWNHRkkvJShsYXJnZW9wR0ZJLyUubW92YWJsZWxpbWl0c0dGSS8lJ2FjY2VudEdGSS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYsNiVRInlGJ0YvRjJGQS1GLDYlUSJ6RidGL0YyLyUrZXhlY3V0YWJsZUdGSUZFRkUvJSVvcGVuR1EiW0YnLyUmY2xvc2VHUSJdRidGQS1JI21uR0YkNiRRIjFGJ0ZFRlxvRkVGRS1GQjYtUSI7RidGRUZHRkpGTEZORlBGUkZURlYvRlpRLDAuMjc3Nzc3OGVtRidGXG9GRQ==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEkZXNwRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYkLUYjNictRjY2Ji1GIzYpLUYsNiVRInhGJ0YvRjItSSNtb0dGJDYtUSIsRicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRkkvJSpzeW1tZXRyaWNHRkkvJShsYXJnZW9wR0ZJLyUubW92YWJsZWxpbWl0c0dGSS8lJ2FjY2VudEdGSS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYsNiVRInlGJ0YvRjJGQS1GLDYlUSJ6RidGL0YyLyUrZXhlY3V0YWJsZUdGSUZFRkUvJSVvcGVuR1EiW0YnLyUmY2xvc2VHUSJdRidGQS1JI21uR0YkNiRRIjJGJ0ZFRlxvRkVGRS1GQjYtUSI7RidGRUZHRkpGTEZORlBGUkZURlYvRlpRLDAuMjc3Nzc3OGVtRidGXG9GRQ==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEkZXNwRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYkLUYjNictRjY2Ji1GIzYpLUYsNiVRInhGJ0YvRjItSSNtb0dGJDYtUSIsRicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRkkvJSpzeW1tZXRyaWNHRkkvJShsYXJnZW9wR0ZJLyUubW92YWJsZWxpbWl0c0dGSS8lJ2FjY2VudEdGSS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYsNiVRInlGJ0YvRjJGQS1GLDYlUSJ6RidGL0YyLyUrZXhlY3V0YWJsZUdGSUZFRkUvJSVvcGVuR1EiW0YnLyUmY2xvc2VHUSJdRidGQS1JI21uR0YkNiRRIjNGJ0ZFRlxvRkVGRS1GQjYtUSI7RidGRUZHRkpGTEZORlBGUkZURlYvRlpRLDAuMjc3Nzc3OGVtRidGXG9GRQ==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEkZXNwRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYkLUYjNiYtRjY2Ji1GIzYoLUYsNiVRInhGJ0YvRjItSSNtb0dGJDYtUSIsRicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRkkvJSpzeW1tZXRyaWNHRkkvJShsYXJnZW9wR0ZJLyUubW92YWJsZWxpbWl0c0dGSS8lJ2FjY2VudEdGSS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYsNiVRInlGJ0YvRjJGQS1GLDYlUSJ6RidGL0YyRkVGRS8lJW9wZW5HUSJbRicvJSZjbG9zZUdRIl1GJ0ZBLUkjbW5HRiQ2JFEiNEYnRkVGRUZFLUZCNi1RIjtGJ0ZFRkdGSkZMRk5GUEZSRlRGVi9GWlEsMC4yNzc3Nzc4ZW1GJy8lK2V4ZWN1dGFibGVHRklGRQ==JSFHLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEkZXNwRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYkLUYjNictRjY2Ji1GIzYnLUYsNiVRInhGJ0YvRjItSSNtb0dGJDYtUSIsRicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRkkvJSpzeW1tZXRyaWNHRkkvJShsYXJnZW9wR0ZJLyUubW92YWJsZWxpbWl0c0dGSS8lJ2FjY2VudEdGSS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYsNiVRInlGJ0YvRjIvJStleGVjdXRhYmxlR0ZJRkVGRS8lJW9wZW5HUSJbRicvJSZjbG9zZUdRIl1GJ0ZBLUkjbW5HRiQ2JFEiMUYnRkVGaW5GRUZFLUZCNi1RIjtGJ0ZFRkdGSkZMRk5GUEZSRlRGVi9GWlEsMC4yNzc3Nzc4ZW1GJ0ZpbkZFLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEkZXNwRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYkLUYjNiYtRjY2Ji1GIzYmLUYsNiVRInhGJ0YvRjItSSNtb0dGJDYtUSIsRicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRkkvJSpzeW1tZXRyaWNHRkkvJShsYXJnZW9wR0ZJLyUubW92YWJsZWxpbWl0c0dGSS8lJ2FjY2VudEdGSS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYsNiVRInlGJ0YvRjJGRUZFLyUlb3BlbkdRIltGJy8lJmNsb3NlR1EiXUYnRkEtSSNtbkdGJDYkUSIyRidGRUZFRkUtRkI2LVEiO0YnRkVGR0ZKRkxGTkZQRlJGVEZWL0ZaUSwwLjI3Nzc3NzhlbUYnLyUrZXhlY3V0YWJsZUdGSUZFLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEkZXNwRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYkLUYjNiYtRjY2Ji1GIzYqLUYsNiVRInhGJ0YvRjItSSNtb0dGJDYtUSIsRicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRkkvJSpzeW1tZXRyaWNHRkkvJShsYXJnZW9wR0ZJLyUubW92YWJsZWxpbWl0c0dGSS8lJ2FjY2VudEdGSS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYsNiVRInlGJ0YvRjJGQS1GLDYlUSJ6RidGL0YyRkEtRiw2JVEid0YnRi9GMkZFRkUvJSVvcGVuR1EiW0YnLyUmY2xvc2VHUSJdRidGQS1JI21uR0YkNiRRIjRGJ0ZFRkVGRS1GQjYtUSI7RidGRUZHRkpGTEZORlBGUkZURlYvRlpRLDAuMjc3Nzc3OGVtRicvJStleGVjdXRhYmxlR0ZJRkU=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEkZXNwRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYkLUYjNictRjY2Ji1GIzYrLUYsNiVRInhGJ0YvRjItSSNtb0dGJDYtUSIsRicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRkkvJSpzeW1tZXRyaWNHRkkvJShsYXJnZW9wR0ZJLyUubW92YWJsZWxpbWl0c0dGSS8lJ2FjY2VudEdGSS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYsNiVRInlGJ0YvRjJGQS1GLDYlUSJ6RidGL0YyRkEtRiw2JVEid0YnRi9GMi8lK2V4ZWN1dGFibGVHRklGRUZFLyUlb3BlbkdRIltGJy8lJmNsb3NlR1EiXUYnRkEtSSNtbkdGJDYkUSIzRidGRUZfb0ZFRkUtRkI2LVEiO0YnRkVGR0ZKRkxGTkZQRlJGVEZWL0ZaUSwwLjI3Nzc3NzhlbUYnRl9vRkU=JSFHLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEkZXNwRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYkLUYjNictRjY2Ji1GIzYtLUYsNiVRInhGJ0YvRjItSSNtb0dGJDYtUSIsRicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRkkvJSpzeW1tZXRyaWNHRkkvJShsYXJnZW9wR0ZJLyUubW92YWJsZWxpbWl0c0dGSS8lJ2FjY2VudEdGSS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYsNiVRInlGJ0YvRjJGQS1GLDYlUSJ6RidGL0YyRkEtRiw2JVEid0YnRi9GMkZBLUYsNiVRInVGJ0YvRjIvJStleGVjdXRhYmxlR0ZJRkVGRS8lJW9wZW5HUSJbRicvJSZjbG9zZUdRIl1GJ0ZBLUkjbW5HRiQ2JFEiNEYnRkVGYm9GRUZFLUZCNi1RIjtGJ0ZFRkdGSkZMRk5GUEZSRlRGVi9GWlEsMC4yNzc3Nzc4ZW1GJ0Zib0ZFLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYqLUkjbW9HRiQ2L1EifkYnLyUlYm9sZEdRJXRydWVGJy8lLG1hdGh2YXJpYW50R1ElYm9sZEYnLyUrZm9udHdlaWdodEdGNC8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGOS8lKXN0cmV0Y2h5R0Y5LyUqc3ltbWV0cmljR0Y5LyUobGFyZ2VvcEdGOS8lLm1vdmFibGVsaW1pdHNHRjkvJSdhY2NlbnRHRjkvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZIRistSSNtaUdGJDYoUSJURidGLy8lJ2l0YWxpY0dGOS8lKnVuZGVybGluZUdGMUYyRjUtRkw2KFEib0YnRi9GT0ZRRjJGNS1GTDYoUT5+Z2VuZXJhdGV+aG9tb2dlbm91c35wcm9kdWN0c0YnRi9GT0ZRRjJGNS8lK2V4ZWN1dGFibGVHRjEvJTBmb250X3N0eWxlX25hbWVHUSkyRH5JbnB1dEYnL0YzUSdub3JtYWxGJw== It is the sum of all productsor r variables selected from n variables ( r\342\211\244 n) allowing repitition. It is a homogenous polynomial of degree r. It contans 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 arguments : list of vaiables L and degree rhp:=proc(V,n)local p,L,M,C,K,N;p:=expand((add(h,h=V))^n);L := [seq(op(i, p), i = 1 .. numbcomb(n+nops(V)-1,nops(V)-1))];M := 2*Vector[row](L);C := [seq(op(1, g), g = M)];K := map(proc (x) options operator, arrow; 1/x end proc, C);N := Vector[row](K);sort(DotProduct(M, N),order=tdeg(V[]));end proc;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JSFH To generate Vandermonde's polynomials argument : list of variables Lvan:=proc(L);sort((-1)^(nops(L)*(nops(L)-1)/2)* Determinant(VandermondeMatrix(L)),order=tdeg(L[])); end proc;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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEnZmFjdG9yRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYkLUYjNiUtRiw2JVEiZ0YnRi9GMi8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0YzUSdub3JtYWxGJ0ZALUkjbW9HRiQ2LVEiO0YnRkAvJSZmZW5jZUdGPy8lKnNlcGFyYXRvckdGMS8lKXN0cmV0Y2h5R0Y/LyUqc3ltbWV0cmljR0Y/LyUobGFyZ2VvcEdGPy8lLm1vdmFibGVsaW1pdHNHRj8vJSdhY2NlbnRHRj8vJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4yNzc3Nzc4ZW1GJ0Y9RkA=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 To generate alternating polynomials A product of Vandermonde polynomial and symmetric 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 To check symmetric polynomials This procedure is based on the following Theorem Let f \342\210\210\342\204\202[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 by G. Then f is symmetric if g depends only on 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 arguments : polynomial f and list of variables V.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=symchk:=proc(f,V)local A,L,GL,B;A:=[seq(a[i],i=1..nops(V))];L:=[seq(esp(V, i), i = 1 .. nops(V))]-A;GL:=Basis(L,tdeg(V[]));B:=NormalForm(f,GL,tdeg(V[]));if is (indets(B)subset convert(A,set)) then true;else false;fi; end proc;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 check alternating 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 is alternating if \316\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 arguments : polynomial f and list of variables VJSFHLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=altchk:=proc(f,V) local A,g,r;g:=van(V);r := NormalForm(f, [g], tdeg(V[]), 'Q');Q;if is(r,0)and is(symchk(Q[], V))then true;else false;fi;end proc;ExamplesLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIzo9RicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkwtRiM2Pi1GIzYqLUklbXN1cEdGJDYlLUYsNiVRInhGJ0YvRjItSSNtbkdGJDYkUSI0RidGOS8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRictRjY2LVExJkludmlzaWJsZVRpbWVzO0YnRjlGO0Y+RkBGQkZERkZGSC9GS1EmMC4wZW1GJy9GTkZeby1GLDYlUSJ5RidGL0YyLyUrZm9yZWdyb3VuZEdRKlswLDAsMjU1XUYnLyUrZXhlY3V0YWJsZUdGPS8lKXJlYWRvbmx5R0YxLyUwZm9udF9zdHlsZV9uYW1lR1EqMkR+T3V0cHV0RidGOS1GNjYtUSgmbWludXM7RidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjIyMjIyMjJlbUYnL0ZORmFwLUYjNiotRlQ2JUZWLUZaNiRRIjNGJ0Y5RmduRmpuLUZUNiVGYG8tRlo2JFEiMkYnRjlGZ25GY29GZm9GaG9Gam9GOS1GNjYtUSIrRidGOUY7Rj5GQEZCRkRGRkZIRmBwRmJwLUYjNiotRlQ2JUZWRlxxRmduRmpuLUZUNiVGYG9GZ3BGZ25GY29GZm9GaG9Gam9GOUZdcC1GIzYqRlZGam4tRlQ2JUZgb0ZZRmduRmNvRmZvRmhvRmpvRjlGXXAtRiM2KkZTRmpuLUYsNiVRInpGJ0YvRjJGY29GZm9GaG9Gam9GOUZfcS1GIzYqRmpxRmpuRl5yRmNvRmZvRmhvRmpvRjlGX3EtRiM2KkZlcEZqbi1GVDYlRl5yRlxxRmduRmNvRmZvRmhvRmpvRjlGXXAtRiM2KkZmcUZqbkZlckZjb0Zmb0Zob0Zqb0Y5Rl1wLUYjNipGZHFGam4tRlQ2JUZeckZncEZnbkZjb0Zmb0Zob0Zqb0Y5Rl9xLUYjNipGanBGam5GW3NGY29GZm9GaG9Gam9GOUZfcS1GIzYqRlZGam4tRlQ2JUZeckZZRmduRmNvRmZvRmhvRmpvRjlGXXAtRiM2KkZgb0ZqbkZhc0Zjb0Zmb0Zob0Zqb0Y5RmNvRmZvRmhvRmpvRjktRjY2LVEiO0YnRjlGOy9GP0YxRkBGQkZERkZGSEZdb0ZNRmZvRjk=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEnYWx0Y2hrRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYkLUYjNigtRiw2JVEiZkYnRi9GMi1JI21vR0YkNi1RIixGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGRS8lKnN5bW1ldHJpY0dGRS8lKGxhcmdlb3BHRkUvJS5tb3ZhYmxlbGltaXRzR0ZFLyUnYWNjZW50R0ZFLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRjY2Ji1GIzYpLUYsNiVRInhGJ0YvRjJGPS1GLDYlUSJ5RidGL0YyRj0tRiw2JVEiekYnRi9GMi8lK2V4ZWN1dGFibGVHRkVGQUZBLyUlb3BlbkdRIltGJy8lJmNsb3NlR1EiXUYnLUYsNiNRIUYnRl9vRkFGQS1GPjYtUSI7RidGQUZDRkZGSEZKRkxGTkZQRlIvRlZRLDAuMjc3Nzc3OGVtRidGX29GQQ==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEnYWx0Y2hrRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYkLUYjNictRiw2JVEicUYnRi9GMi1JI21vR0YkNi1RIixGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGRS8lKnN5bW1ldHJpY0dGRS8lKGxhcmdlb3BHRkUvJS5tb3ZhYmxlbGltaXRzR0ZFLyUnYWNjZW50R0ZFLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRjY2Ji1GIzYoLUYsNiVRInhGJ0YvRjJGPS1GLDYlUSJ5RidGL0YyRj0tRiw2JVEiekYnRi9GMkZBRkEvJSVvcGVuR1EiW0YnLyUmY2xvc2VHUSJdRictRiw2I1EhRidGQUZBLUY+Ni1RIjtGJ0ZBRkNGRkZIRkpGTEZORlBGUi9GVlEsMC4yNzc3Nzc4ZW1GJy8lK2V4ZWN1dGFibGVHRkVGQQ==JSFHLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEnYWx0Y2hrRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYkLUYjNistSSVtc3VwR0YkNiUtRiw2JVEieEYnRi9GMi1GIzYlLUkjbW5HRiQ2JFEiMkYnL0YzUSdub3JtYWxGJ0YvRjIvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnLUkjbW9HRiQ2LVEiK0YnRkYvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRlEvJSlzdHJldGNoeUdGUS8lKnN5bW1ldHJpY0dGUS8lKGxhcmdlb3BHRlEvJS5tb3ZhYmxlbGltaXRzR0ZRLyUnYWNjZW50R0ZRLyUnbHNwYWNlR1EsMC4yMjIyMjIyZW1GJy8lJ3JzcGFjZUdGam4tRjs2JS1GLDYlUSJ5RidGL0YyLUYjNiUtRkM2JFEiM0YnRkZGL0YyRkhGSy1GOzYlLUYsNiVRInpGJ0YvRjItRiM2JS1GQzYkUSI0RidGRkYvRjJGSC1GTDYtUSIsRidGRkZPL0ZTRjFGVEZWRlhGWkZmbi9GaW5RJjAuMGVtRicvRlxvUSwwLjMzMzMzMzNlbUYnLUY2NiYtRiM2KEY9RmFwRl9vRmFwRmlvRkZGRi8lJW9wZW5HUSJbRicvJSZjbG9zZUdRIl1GJy1GLDYjUSFGJ0ZGRkYtRkw2LVEiO0YnRkZGT0ZkcEZURlZGWEZaRmZuRmVwL0Zcb1EsMC4yNzc3Nzc4ZW1GJy8lK2V4ZWN1dGFibGVHRlFGRg==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbW9HRiQ2LVEifkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDLUkjbWlHRiQ2JVEnYWx0Y2hrRicvJSdpdGFsaWNHUSV0cnVlRicvRjBRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYkLUYjNiotSSNtbkdGJDYkUSIyRidGLy1GLDYtUScmc2RvdDtGJ0YvRjJGNUY3RjlGO0Y9Rj9GQUZELUklbXN1cEdGJDYlLUZQNiQtRiM2Ji1GRzYlUSJ4RidGSkZNLUYsNi1RKCZtaW51cztGJ0YvRjJGNUY3RjlGO0Y9Rj8vRkJRLDAuMjIyMjIyMmVtRicvRkVGY28tRkc2JVEieUYnRkpGTUYvRi8tRiM2JS1GVTYkUSIzRidGL0ZKRk0vJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnLUYsNi1RIixGJ0YvRjIvRjZGTEY3RjlGO0Y9Rj9GQS9GRVEsMC4zMzMzMzMzZW1GJy1GUDYmLUYjNiZGXG9GYHBGZW9GL0YvLyUlb3BlbkdRIltGJy8lJmNsb3NlR1EiXUYnRmBwLUZQNiYtRiM2Ji1GRzYlUSJhRidGSkZNRmBwLUZHNiVRImJGJ0ZKRk1GL0YvRmpwRl1xRi9GLy1GLDYtUSI7RidGL0YyRmNwRjdGOUY7Rj1GP0ZBL0ZFUSwwLjI3Nzc3NzhlbUYnLyUrZXhlY3V0YWJsZUdGNEYvLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEnYWx0Y2hrRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYkLUYjNiktRiw2JVEieEYnRi9GMi1JI21vR0YkNi1RKCZtaW51cztGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkUvJSlzdHJldGNoeUdGRS8lKnN5bW1ldHJpY0dGRS8lKGxhcmdlb3BHRkUvJS5tb3ZhYmxlbGltaXRzR0ZFLyUnYWNjZW50R0ZFLyUnbHNwYWNlR1EsMC4yMjIyMjIyZW1GJy8lJ3JzcGFjZUdGVC1JJW1zdXBHRiQ2JS1GLDYlUSJ5RidGL0YyLUYjNiQtRiw2I1EhRidGQS8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRictRj42LVEiLEYnRkFGQy9GR0YxRkhGSkZMRk5GUC9GU1EmMC4wZW1GJy9GVlEsMC4zMzMzMzMzZW1GJy1GNjYmLUYjNiZGOkZfb0ZaRkFGQS8lJW9wZW5HUSJbRicvJSZjbG9zZUdRIl1GJ0ZpbkZBRkEtRj42LVEiO0YnRkFGQ0Zib0ZIRkpGTEZORlBGY28vRlZRLDAuMjc3Nzc3OGVtRicvJStleGVjdXRhYmxlR0ZFRkE=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEnYWx0Y2hrRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYkLUYjNiYtSSNtbkdGJDYkUSIxRicvRjNRJ25vcm1hbEYnLUkjbW9HRiQ2LVEiLEYnRj4vJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGRi8lKnN5bW1ldHJpY0dGRi8lKGxhcmdlb3BHRkYvJS5tb3ZhYmxlbGltaXRzR0ZGLyUnYWNjZW50R0ZGLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRjY2Ji1GIzYmLUYsNiVRInhGJ0YvRjJGQC1GLDYlUSJ5RidGL0YyRj5GPi8lJW9wZW5HUSJbRicvJSZjbG9zZUdRIl1GJ0Y+Rj4tRkE2LVEiO0YnRj5GREZHRklGS0ZNRk9GUUZTL0ZXUSwwLjI3Nzc3NzhlbUYnLyUrZXhlY3V0YWJsZUdGRkY+LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEnYWx0Y2hrRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYkLUYjNigtSSVtc3VwR0YkNiUtRjY2JC1GIzY+LUY7NiUtRiw2JVEieEYnRi9GMi1JI21uR0YkNiRRIjJGJy9GM1Enbm9ybWFsRicvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnLUkjbW9HRiQ2LVExJkludmlzaWJsZVRpbWVzO0YnRkovJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRlUvJSlzdHJldGNoeUdGVS8lKnN5bW1ldHJpY0dGVS8lKGxhcmdlb3BHRlUvJS5tb3ZhYmxlbGltaXRzR0ZVLyUnYWNjZW50R0ZVLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGXm8tRiw2JVEieUYnRi9GMi1GUDYtUSgmbWludXM7RidGSkZTRlZGWEZaRmZuRmhuRmpuL0Zdb1EsMC4yMjIyMjIyZW1GJy9GYG9GaG9GQ0ZPLUY7NiVGYW9GRkZMRmRvRkFGTy1GLDYlUSJ6RidGL0YyLUZQNi1RIitGJ0ZKRlNGVkZYRlpGZm5GaG5Gam5GZ29GaW9Gam9GT0ZccEZfcEZDRk8tRjs2JUZccEZGRkxGZG9GYW9GT0ZicC8lK2ZvcmVncm91bmRHUSpbMCwwLDI1NV1GJy8lK2V4ZWN1dGFibGVHRlUvJSlyZWFkb25seUdGMS8lMGZvbnRfc3R5bGVfbmFtZUdRKjJEfk91dHB1dEYnRkpGSi1GIzYlLUZHNiRRIjNGJ0ZKRi9GMkZMLUZQNi1RIixGJ0ZKRlMvRldGMUZYRlpGZm5GaG5Gam5GXG8vRmBvUSwwLjMzMzMzMzNlbUYnLUY2NiYtRiM2KUZDRmNxRmFvRmNxRlxwRmdwRkpGSi8lJW9wZW5HUSJbRicvJSZjbG9zZUdRIl1GJy1GLDYjUSFGJ0ZncEZKRkotRlA2LVEiO0YnRkpGU0ZmcUZYRlpGZm5GaG5Gam5GXG8vRmBvUSwwLjI3Nzc3NzhlbUYnRmdwRko=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=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 express a symmetric polynomial in terms of elementary symmetric polynomials arguments : symmetric polynomial f, list of variables V and list of constants A representing elementary symmetric polynomials.rep:=proc(f,V,A)local L,G,GL;L:=[seq(esp(V, i), i = 1 .. nops(V))]-A;GL:=Basis(L,tdeg(V[]));NormalForm(f,GL,tdeg(V[]));end proc;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To find the discriminant of a univariate polynomial(equation) The discriminant of 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 argument : list of the coefficients C = [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disc:=proc(C) local n,E,V,L,f,GL;n:=nops(C);E:=[seq((-1)^(i-1)*C[i]/C[1],i=2..n)];V:=[seq(x[i],i=1..n-1)];L:=[seq(esp(V,i),i=1..n-1)]-E;f:=(C[1])^(2*n-4)*(van(V))^2;GL:=Basis(L,tdeg(V[]));NormalForm(f,GL,tdeg(V[]));end proc; ExamplesLinear equation ax+b=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 equation 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Cubic equation 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Cubic equation 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Quartic equation 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 Vandermonde polynomial = 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JSFH