### The Cayley–Menger determinant is irreducible for $n\ge 3$.

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Let Δ denote the discriminant of the generic binary -ic. We show that for d ≥ 3, the Jacobian ideal of Δ is perfect of height 2. Moreover we describe its SL-equivariant minimal resolution and the associated differential equations satisfied by Δ. A similar result is proved for the resultant of two forms of orders , whenever ≥ -1. If Φ denotes the locus of binary forms with total root multiplicity ≥ -n, then we show that the ideal of Φ is also perfect, and we construct a covariant which characterizes...

We present bounds for the degree and the height of the polynomials arising in some problems in effective algebraic geometry including the implicitization of rational maps and the effective Nullstellensatz over a variety. Our treatment is based on arithmetic intersection theory in products of projective spaces and extends to the arithmetic setting constructions and results due to Jelonek. A key role is played by the notion of canonical mixed height of a multiprojective variety. We study this notion...

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