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Extra info for Applying Informed Coding and Embedding to Design a Robust, High Capacity Watermark

Example text

Sekigawa such that the inequality |tμ | < |t| and the equations in (5) hold. This contradicts the assumption that f˜ is a nearest polynomial. Therefore, the determinant is not 0 in a neighborhood U ⊂ I of α. Hence, we can write tμ and t as functions of x, that is, tμ (x) and t(x). If |t(x)| does not have a minimum at x = α, by moving β in I from α, we can decrease |t(β)| such that the condition |tμ (β)| < |t(β)| and we obtain a polynomial g(x) = f (x) + ni=1 bi ei (x) (bi ∈ R) such that g(β) = 0 and f − g ∞ < f − f˜ ∞ .

Substituting b1 = 0 into (14) gives k2 = 13 (b32 )1/3 + b22 /3(b32 )1/3 + b2 /3 For b2 > 0 this gives k2 = b2 , while for b2 < 0 it simplifies to k2 = 0. For b2 = 0, the system should report a divide by zero error. Thus for b2 = 0, (13) and (14) work even for b1 = 0, although it should be noted that the position of the minimum, −b1 /2k2 , will give a divide by zero error for all b2 < 0. It is important to remember in this discussion that the mathematical properties of (13) and (14). Thus, the fact that it is possible to obtain the correct result for b1 = b2 = 0 by taking limits is not relevant; what is relevant is how a CAS will manipulate the expressions.

Xn , U ] + U is generated by a non-constant polynomial Pk . If for all k ∈ {1, . . , n − j + 1}, α does not belong to the leading coefficient of Pk seen as a univariate polynomial −1 in Xk , πn−j+2 (α) has a zero-dimensional intersection A with the variety defined by Δn−j+1 . This intersection lies in C. Remark now that C is equi-dimensional since I is equi-dimensional, so that the points in A are not isolated in C. Theorem 3. Let f be a polynomial in Q[X1 , . . , Xn ]. Suppose that for all i ∈ {1, .

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Applying Informed Coding and Embedding to Design a Robust, High Capacity Watermark by Miller, Doerr, Cox

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