Sparse squares of polynomials

Sparse squares of polynomials

We answer a question left open in an article of Coppersmith and Davenport which proved the existence of polynomials whose powers are sparse, and in particular polynomials whose squares are sparse (i.e., the square has fewer terms than the original polynomial). They exhibit some polynomials of degree 12 having sparse squares, and ask whether there are any lower degree complete polynomials with t...

Polynomials than Sums of Squares

We investigate the quantitative relationship between nonnegative polynomials and sums of squares of polynomials. We show that if the degree is fixed and the number of variables grows then there are significantly more nonnegative polynomials than sums of squares. More specifically, we take compact bases of the cone of nonnegative polynomials and the cone of sums of squares and derive bounds for ...

Factors of Sparse Polynomials are Sparse

Abstract This paper was removed due to an error in the proof (Claim 4.12 as stated is not true). The authors would like to thank Ilya Volkovich for pointing out a counterexample to this papers main result in positive characteristic: If F is a field with prime characteristic p, then the polynomial xp1 + x p 2 + . . . + x p n has the following factor: (x1 + x2 + . . . + xn) p−1, which has sparsit...

Sudoku Squares and Chromatic Polynomials

T he Sudoku puzzle has become a very popular puzzle that many newspapers carry as a daily feature. The puzzle consists of a 9×9 grid in which some of the entries of the grid have a number from 1 to 9. One is then required to complete the grid in such a way that every row, every column, and every one of the nine 3× 3 sub-grids contain the digits from 1 to 9 exactly once. The sub-grids are shown ...

Probabilistic Algorithms Sparse Polynomials