query: If a polynomial $f(X,Y)$ vanishes on all $(x,y)$, can we conclude $f(X,Y)=0$?
positive_passages.docid: 27675
positive_passages.text: Polynomials vanishing on an infinite set
negative_passages.docid: 151519
negative_passages.text: If $f(f(x))=x^2-x+1$, what is $f(0)$?
negative_passages.docid: 121663
negative_passages.text: If $a$ is a zero divisor of $\mathbb{F}$, then does $a$ fail to have a multiplicative inverse?
negative_passages.docid: 71799
negative_passages.text: A complex polynomial with partial derivatives equal to zero is constant.
negative_passages.docid: 53012
negative_passages.text: Irreducibility of a polynomial if it has no root (Capelli)
negative_passages.docid: 128043
negative_passages.text: $A^n=0$ for $A$ nilpotent without the use of minimal polynomial
negative_passages.docid: 67891
negative_passages.text: Why are there no discrete zero sets of a polynomial in two complex variables?
negative_passages.docid: 89473
negative_passages.text: Uniqueness of minimal polynomial: $f(x)$ divides $g(x)$
negative_passages.docid: 103932
negative_passages.text: How can I prove that a polynomial with degree $n$ is continuous everywhere in $\mathbb{R}$ using definitions?
negative_passages.docid: 40502
negative_passages.text: The Degree of Zero Polynomial.
negative_passages.docid: 70249
negative_passages.text: Irreducibility of Polynomials in $k[x,y]$
negative_passages.docid: 87743
negative_passages.text: Is a function whose derivative vanishes at rationals constant?
negative_passages.docid: 103340
negative_passages.text: Show that this entire function is polynomial.
negative_passages.docid: 33247
negative_passages.text: Does $f(0)=0$ and $\left|f^\prime(x)\right|\leq\left|f(x)\right|$ imply $f(x)=0$?
negative_passages.docid: 39179
negative_passages.text: Why is $(0, 0)$ not a minimum of $f(x, y) = (y-3x^2)(y-x^2)$?
negative_passages.docid: 42265
negative_passages.text: A proof that $\displaystyle\sum_{i=1}^{n}P''(r_i)/P'(r_i)=0$ for a degree $n$ polynomial
negative_passages.docid: 121304
negative_passages.text: Is the set of polynomials of degree less than or equal to $n$ closed?