By Adamek J., Trnkova V.

From "Theory and functions of Categories", Vol. 14, No. 18, 2005, pp. 424-450.

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**Example text**

The procedure has two obligatory input parameters: polynomial g(Q) and a name of the ﬁle for storing coordinates of the carrier points in the format of program qconvex. An optional parameter is a list of names of variables for which it is required to construct the support of the polynomial. The procedure uses auxiliary procedure PGsupp. PGgetnormals gets information on the Newton polyhedron, its faces, and normals to them and converts it into a list of normal vectors with integer coeﬃcients. The procedure has one obligatory parameter – a name of the ﬁle with results of operation of program qconvex – and returns list of support planes of the Newton polyhedron determined by the normal vector and shift.

K is maximum}, giving the E2 function, deﬁned similarly as above. Example 1. E1 (42) = (3, 0, 2, 1, 3, 0, 2, 0). The corresponding power raising computation sequence is u → u3 → u6 → u7 → u21 → u42 . E2 (42) coincides instead with binary decomposition. 3 the diﬀerent decompositions of exponents up to 100 given by the E1 (·) and E2 (·) functions, only when they diﬀer. For brevity, we omit all zero entries. We observe that E1 tends to use cubes more often and possibly earlier (with smaller operands), while E2 uses fewer cubes, and later (larger operands).

Moreover, the polynomial f (μ) has the root of multiplicity 3 along the family P1 . Direct computations show that there are two singular points of order 2 Q0 = (0, 0, 0) and Q1 = (−2, 2, 2) in the family P1 . At the point Q1 , the polynomial f (μ) has the root of multiplicity 4. 2 Expansion of the Family P2 of Singular Points In case (2), we make the substitution x1 = x4 , y1 = y4 + x24 /8, z1 = z4 . (18) (1) The faces adjacent to the edge Γ12 , corresponding to polynomial (9), have normals N12 = (1, 2, 1) and N17 = (0, 0, −1).

### Birkhoff's variety theorem with and without free algebras by Adamek J., Trnkova V.

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