We present lazy and forgetful algorithms for adding, multiplying and dividing multivariate polynomials. The lazy property allows us to compute the i-th term of a polynomial without doing the work required to compute all the terms. The forgetful property allows us to forget earlier terms that have been computed to save space. For example, given polynomials A,B,C,D,E we can compute the exact quot...

In this paper, we revisit the greatest common right divisor (GCRD) extraction from a set of polynomial matrices , with coefficients in generic field and column dimension . We give necessary sufficient conditions for matrix to be GCRD using Smith normal form compound obtained by concatenating vertically, where also describe complete degrees freedom solution link it Hermite then an algorithm cons...

The formula mentioned in the title is proved. Introduction Let S, T be complete nonsingular surfaces over an algebraically closed field k of any characteristic, and let h : T → S be a finite separable morphism of degree n. We establish a formula that expresses the Euler characteristic (understood as the degree of the second Chern class ∫ c2,T ) of T via the Euler characteristic of S and some lo...

We study the behavior of the greatest common divisor of a k − 1 and b k − 1, where a, b are fixed integers or polynomials, and k varies. In the integer case, we conjecture that when a and b are multiplicatively independent and in addition a− 1 and b − 1 are coprime, then a k − 1 and b k − 1 are coprime infinitely often. In the polynomial case, we prove a strong version of this conjecture. To do...

Euclid’s algorithm gives the greatest common divisor (gcd) of two integers, gcd(a, b) = max{d ∈ Z | d|a, d|b} If for simplicity we define gcd(0, 0) = 0, we have a function gcd : Z× Z −→ N with the following properties: Lemma 1 For any a, b, c, q ∈ Z we have: (i) gcd(a, b) = gcd(b, a). (ii) gcd(a,−b) = gcd(a, b). (iii) gcd(a, 0) = |a|. (iv) gcd(a− qb, b) = gcd(a, b). Proof. Trivial; for (iv) use...

We discuss computation of approximate Gröbner bases at high but finite precision. We show how this can be used to deduce exact results for various applications. Examples include implicitizing surfaces, finding multivariate polynomial greatest common divisors and factorizations over the rational and complex number fields. This is an extended version of a paper for SYNASC 2010: Proceedings of the...

Let G be a finite group acting faithfully on an irreducible non-singular projective curve defined over an algebraically closed field F . Does every G-invariant divisor class contain a Ginvariant divisor? The answer depends only on G and not on the curve. We answer the same question for degree 0 divisor (classes). We investigate the question for cycles on varieties.

We define a Deligne-Mumford stack XD,r which depends on a scheme X , an effective Cartier divisor D ⊂ X , and a positive integer r . Then we show that the Abramovich-Vistoli moduli stack of stable maps into XD,r provides compactifications of the locally closed substacks of M̄g,n(X,β) corresponding to relative stable maps. We also state an enumerative result counting rational plane curves with ta...

We study the complexity of expressing the greatest common divisor of n positive numbers as a linear combination of the numbers. We prove the NP-completeness of finding an optimal set of multipliers with respect to either the L0 metric or the L∞ norm. We present and analyze a new method for expressing the gcd of n numbers as their linear combination and give an upper bound on the size of the lar...