Bernshtěın’s second theorem
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چکیده
Solutions at infinity are solutions with z0 = 0 and not all other coordinates equal to zero. We see that only z 1z 2 2 = 0 remains, or equivalently we find [0 : 1 : 0] and [0 : 0 : 1] representing two distinct solutions at infinity, but each with multiplicity two. This is an unsatisfactory result, as we expected to find two, not four solutions at infinity. The projective transformation defined by x1 = z1z −1 0 and x2 = z2z −1 0 assumes one hyperplane at infinity, defined by z0 = 0. The exponents of z0 in this transformation −1 and −1 form the inner normal (−1,−1) to the edge of the Newton polygon on which the highest degree monomials are supported. Let us look at the edge with inner normal (−2,+1) and use the coordinate transformation x1 = z1z −2 0 and x2 = z2z +1 0 . After this substitution and multiplication by z 0 we then find:
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