We focus on rational solutions or nearly-feasible that serve as certificates of feasibility for polynomial optimization problems. show that, under some separability conditions, certain cubic polynomially constrained sets admit solutions. However, we in other cases it is NP Hard to detect if exist they any reasonable size. extend this idea various settings including near feasible, but super opti...

In this paper, an original reduction algorithm for solving simultaneous multivariate polynomial equations is presented. The exponential in complexity, but the well-known algorithms, such as extended Euclidean and Buchberger's algorithm, are superexponential. superexponential complexity of algorithms due to their not being "minimal" a certain sense. produces Groebner basis. proposed achieves req...

Abstract For a polynomial progression $$ \begin{align*}(x,\; x+P_1(y),\ldots,\; x+P_{t}(y)),\end{align*} we define four notions of complexity: Host–Kra complexity, Weyl true complexity and algebraic complexity. The first two describe the smallest characteristic factor progression, third refers to smallest-degree Gowers norm controlling fourth concerns relations between terms progressions. We co...

The true complexity of a polynomial progression in finite fields corresponds to the smallest-degree Gowers norm that controls counting operator over large characteristic. We give conjecture relates algebraic relations between terms progression, and we prove it for number progressions, including $x,\; x+y,\; x+y^2,\; x+y+y^2$ x+2y,\; x+y^2$. As corollary, an asymptotic count certain progressions...