Related papers: Polynomial Calculus sizes over the Boolean and Fou…

Let $K$ be a field, $R=K[x, y]$ the polynomial ring and $\mathcal{M}(K)$ the set of all pairs of square matrices of the same size over $K.$ Pairs $P_1=(A_1,B_1)$ and $P_2=(A_2,B_2)$ from $\mathcal{M}(K)$ are called similar if…

We obtain a nontrivial upper bound for almost all elements of the sequences of real numbers which are multiplicative and at the prime indices are distributed according to the Sato--Tate density. Examples of such sequences come from…

In this paper, we investigate large values of Dirichlet polynomials with multiplicative coefficients $\sum_{n\le N}f(n)n^{it}$, where $1\ll t\le T$ for large $T$. We prove an improved Omega result in the region $\exp((\log…

We study a random polynomial of degree $n$ over the finite field $\mathbb{F}_q$, where the coefficients are independent and identically distributed and uniformly chosen from the squares in $\mathbb{F}_q$. Our main result demonstrates that…

We generalize an approach from a 1960 paper by Ljunggren, leading to a practical algorithm that determines the set of $N > \operatorname{deg}(c) + \operatorname{deg}(d)$ such that the polynomial $$f_N(x) = x^N c(x^{-1}) + d(x)$$ is…

In this paper, we uncover a new uncertainty principle that governs the complexity of Boolean functions. This principle manifests as a fundamental trade-off between two central measures of complexity: a combinatorial complexity of its…

Suppose that $f: \bR^n\to\bR^n$ is a mapping of $K$-bounded $p$-mean distortion for some $p>n-1$. We prove the equivalence of the following properties of $f$: doubling condition for $J(x,f)$ over big balls centered at origin, boundedness of…

For a $t$-nomial $f(x) = \sum_{i = 1}^t c_i x^{a_i} \in \mathbb{F}_q[x]$, we show that the number of distinct, nonzero roots of $f$ is bounded above by $2 (q-1)^{1-\varepsilon} C^\varepsilon$, where $\varepsilon = 1/(t-1)$ and $C$ is the…

We prove that the Fourier dimension of any Boolean function with Fourier sparsity $s$ is at most $O\left(s^{2/3}\right)$. Our proof method yields an improved bound of $\widetilde{O}(\sqrt{s})$ assuming a conjecture of…

We answer Totik's question on weighted Bernstein's inequalities showing that $$ \|T_n'\|_{L_p(\omega)} \le C(p,\omega)\, {n}\,\|T_n\|_{L_p(\omega)},\qquad 0<p\le \infty, $$ holds for all trigonometric polynomials $T_n$ and certain…

We prove lower bounds for the Minimum Circuit Size Problem (MCSP) in the Sum-of-Squares (SoS) proof system. Our main result is that for every Boolean function $f: \{0,1\}^n \rightarrow \{0,1\}$, SoS requires degree $\Omega(s^{1-\epsilon})$…

We significantly strengthen and generalize the theorem lifting Nullstellensatz degree to monotone span program size by Pitassi and Robere (2018) so that it works for any gadget with high enough rank, in particular, for useful gadgets such…

In this paper, we consider bounded width circuits and nondeterministic circuits in three somewhat new directions. In the first part of this paper, we mainly consider bounded width circuits. The main purpose of this part is to prove that…

Using the theory of Newton Polygons, we formulate a simple criterion for the Galois group of a polynomial to be ``large.'' For a fixed $\alpha \in \Q - \Z_{<0}$, Filaseta and Lam have shown that the $n$th degree Generalized Laguerre…

We have found the most general extension of the celebrated Sauer, Perles and Shelah, Vapnik and Chervonenkis result from 0-1 sequences to $k$-ary codes still giving a polynomial bound. Let $\mathcal{C}\subseteq \{0,1,..., k-1}^n$ be a…

We use Grothendieck's dessins d'enfant to show that if $P$ and $Q$ are two real polynomials, any real function of the form $x^\alpha(1-x)^{\beta} P - Q$, has at most $\deg P +\deg Q + 2$ roots in the interval $]0,~1[$. As a consequence, we…

Harvey Friedman, in his remarkable paper Finite functions and the necessary use of large cardinals, Ann. Math. 148:803-893, 1998 and in a technical report, Applications of large cardinals to graph theory, Ohio State University, 1997,…

Sensitivity conjecture is a longstanding and fundamental open problem in the area of complexity measures of Boolean functions and decision tree complexity. The conjecture postulates that the maximum sensitivity of a Boolean function is…

Let $F(x)=(f_1(x), \dots, f_m(x))$ be such that $1, f_1, \dots, f_m$ are linearly independent polynomials with real coefficients. Based on ideas of Bachoc, DeCorte, Oliveira and Vallentin in combination with estimating certain oscillatory…

Let $k_i\ (i=1,2,\ldots,t)$ be natural numbers with $k_1>k_2>\cdots>k_t>0$, $k_1\geq 2$ and $t<k_1.$ Given real numbers $\alpha_{ji}\ (1\leq j\leq t,\ 1\leq i\leq s)$, we consider polynomials of the shape…