This aim of this Maple worksheet is to help the reading of Section 3 of the tutorial paper "DDE-SOLVER: A Maple Package For Discrete Differential Equations"
<Text-field style="Heading 1" layout="Heading 1"><Font size="16">ALGORITHM "DUPLICATION"</Font> <Font size="16">(detailed example)</Font></Text-field> 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<Text-field style="Heading 1" layout="Heading 1"><Font size="16">ALGORITHM "ELIMINATION" (detailed example)</Font></Text-field> 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LDYpUSRuZXdGJ0ZqdEZddUZfdUYvRmF1RmN1Rlt3LUYsNilRKXZhcmlhYmxlRidGanRGXXVGX3VGL0ZhdUZjdS1GNjYyUSIuRidGanRGXXVGX3VGYXVGY3VGOUY7Rj1GP0ZBRkNGRUZRRlNGTEZdekYy 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LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJSR0YoKiYsKiooIiM7IiIiSSJ0R0YoRjIpSSN6MEdGKCIiI0YyRjIqKCIiKUYyRjNGMkY1RjIhIiJGM0YyRjFGOUYyLDQqKCIjIilGMilGM0Y2RjIpRjUiIiRGMkYyKihGPEYyRj1GMkY0RjJGOSooIiNGRjJGPUYyRjVGMkYyKigiIz1GMkYzRjJGNEYyRjIqJkY/RjJGPUYyRjkqKCIjbUYyRjNGMkY1RjJGOSomIiNaRjJGM0YyRjJGNUYyRjJGOUYyNyNGLg==
<Text-field style="Heading 1" layout="Heading 1"><Font size="16">ALGORITHM "GEOMETRY" (detailed example)</Font></Text-field> Our first goal is to compute the matrix Mz1 of the multiplication-by-z1 in Q(t, z0)[m, x, u, z1] / (tS), where (tS) is the ideal generated by the polynomials in S, seen as elements of Q(t, z0)[m, x, u, z1]. 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 We then compute the characteristic polynomial xi_z1 of Mz1, whose eigenvalues are exactly the {t, z0}-coordinates of the points in V(ttS), where ttS is now seen as the ideal generated by the polynomials in S, seen as elements of Q[m, x, u, z1, z0, t]. 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEmZGVub21GJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkobWZlbmNlZEdGJDYkLUYjNiUtRiw2JVEmeGlfejFGJ0YvRjIvJStleGVjdXRhYmxlR0YxRjJGMkY9RjI= LCQqJEkidEc2IiIiJiIkSCg= As we are interested in the couples of distinct points in V(ttS) that share the same {t, z0, z1}-coordinates, the polynomial chi_z1 must have, at {t, z0} fixed, a double root in z1. Thus, we compute the discriminant of chi_z1 and finally find an annihilating polynomial fo F(t, 1) in its factors. 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 LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJSR0YoLCQqLEkidEdGKCIjOSwqKiZGMCIiIkkjejBHRigiIiMiIzsqJkYwRjRGNUY0ISIpRjBGNCEjO0Y0RjYsJkY4RjRGNEY0IiM3LDQqJkYwRjZGNSIiJCIjIikqJkYwRjZGNUY2ISMiKSomRjBGNkY1RjQiI0ZGMyIjPSokRjBGNiEiJEY4ISNtRjAiI1pGNUY0ISIiRjRGNiwuRkEiIipGQyEiJ0ZGRjRGOEZNRjAhIzlGNEY0Rj8iLGsvZklBIjcjRi4=
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW9HRiQ2LlEifkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDLyUpZXF1YWxpdHlHRjQvJStleGVjdXRhYmxlR0Y0Ri8=