I have implemented a class of polynomials (and matrices). I checked the performance of all the methods in a separate sandbox project. After that, I split the class into 2 files. .h and .cpp. I tried to use the methods of the class and it turned out that the lnk2019 error occurs in each one. Earlier, I was able to solve a similar error by adding before the class description (inside .h)lines:

template<typename P> class polynomial;
template<typename P> std::ostream& operator<<(std::ostream& out, const polynomial<P>& plnm);
template<typename P> std::istream& operator>> (std::istream& in, polynomial<P>& plnm);

Adding the lines above did not fix the error.

content .h
#include<iostream>
#include <sstream>
template<typename P> class polynomial;
template<typename P> std::ostream& operator<<(std::ostream& out, const polynomial<P>& plnm);
template<typename P> std::istream& operator>> (std::istream& in, polynomial<P>& plnm);

template<typename P> class polynomial
{
public:

    P* ptr;
    uint64_t deg;
    polynomial() :polynomial(0) {};
    polynomial(uint64_t size);
    ~polynomial();
    polynomial<P> operator+(const polynomial<P>& other)const;//addition of polynomials
    polynomial<P> operator-(const polynomial<P>& other)const;
    polynomial<P> operator/(const polynomial<P>& other)const;//binary division of a polynomial by a polynomial
    polynomial<P> operator%(const polynomial<P>& other)const;//the remainder of the division of a polynomial by a polynomial
    polynomial<P> operator*(const polynomial<P>& other)const;//binary polynomial multiplication 
    polynomial<P> operator*(const P& other)const; //binary polynomial multiplication by an element from the field

    friend std::ostream& operator<<<>(std::ostream& out, const polynomial<P>& plnm); // overloading the output operator
    friend std::istream& operator>><>(std::istream& in, polynomial<P>& plnm); // overloading the input operator

    polynomial<P> operator>>(const uint64_t power)const;//coefficient shift (increase). or (plnm*x^n)
    polynomial<P> operator<<(const uint64_t power)const;//coefficient shift(decrease). or the whole part of (plnm* x^(-n))

    //P& operator[](uint64_t index);//the access operator to the polynomial coefficient

    polynomial<P>& operator=(const polynomial<P>& other);

    bool operator==(const polynomial<P>& other)const;// 
    bool operator!=(const polynomial<P>& other)const;
};

content main

int main() {

    polynomial<double> a(2);
    
    std::cin >> a;
    std::cout << "a = n" <<a;

    system("pause");
    return 0;
}

an error occurs with each method, so for brevity, I wrote the simplest option in the main one.

part of the contents of the .cpp file
template<typename P>polynomial<P> polynomial<P>::operator>>(const uint64_t power)const
{
    polynomial result(deg + power);
    for (uint64_t i = 0; i < deg; i++)
    {
        result.ptr[i + power] = ptr[i];
    }
    return result;
}

template<typename P>polynomial<P> polynomial<P>::operator<<(const uint64_t power)const
{
    polynomial result(deg - power);
    for (uint64_t i = 0; i <= (deg - power); i++)
    {
        result.ptr[i] = ptr[i + power];
    }
    return result;
}

I assume that the problem may be related to the fact that the polynomial<T> that I declare in the main is not considered a template. This was the case with the operators >> and <<. But I fixed it with three lines, but the compiler seems to think differently.